Series

Problem 1

Determine if the series converges or diverges: $\sum_{n=1}^{\infty}\left[\left(\frac{6}{7}\right)^{n}-\frac{3}{2^{n}}\right]$ . If it converges, find the sum.

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Problem 2

Find the sum of the series $\sum_{n=1}^{\infty} \frac{1}{n(n+2)}$ .

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Problem 3

Determine if the series are convergent or divergent:
(I) $\sum_{n=1}^{\infty}\left(\frac{n+3}{2 n-5}\right)^{n}$
(II) $\sum_{n=1}^{\infty} \sqrt[4]{\frac{2}{n^{3}}}$
(III) $\sum_{n=1}^{\infty} \frac{2^{n}}{n ! \sqrt{n+1}}$
Explain your reasoning.

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Problem 4

Determine if these series converge or diverge, explaining your reasoning: (I) $\sum_{n=1}^{\infty}\left(\frac{n+3}{2 n-5}\right)^{n}$ , (II) $\sum_{n=1}^{\infty} \sqrt[4]{\frac{2}{n^{3}}}$ , (III) $\sum_{n=1}^{\infty} \frac{2^{n}}{n ! \sqrt{n+1}}$ .

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Problem 5

Determine if these series are absolutely convergent, conditionally convergent, or divergent:
(I) $\sum_{n=1}^{\infty}\left(\frac{\pi}{2}\right)^{n}$
(II) $\sum_{n=1}^{\infty} \frac{(-1)^{n-1} e}{\sqrt{n}}$
(III) $\sum_{n=1}^{\infty} \frac{1}{\sqrt{n}(\sqrt{n}+1)^{3}}$

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Problem 6

Calculate the total of sales volumes: 2,525 million, 4,716 million, 7,241 million, and 9,096 million.

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Problem 7

What is the sum of the series $-2-3-4-5-\cdots-101$ ?

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Problem 8

Express the polynomial $\frac{5}{2} x^{3}+3 x^{4}+\frac{7}{2} x^{5}+4 x^{6}$ in sigma notation. Which option is correct?

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Problem 9

Find the sum of the series: $1 + 2 + 3 + \ldots + 304$ .

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Problem 10

计算从 $\left(\frac{1}{3}+\frac{2}{3}\right)$ 到 $\left(\frac{1}{100}+\cdots+\frac{99}{100}\right)$ 的总和。

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Problem 11

Find the sum of the first 31 terms of an arithmetic series where $t_{1}=-9$ and $t_{2}=9$ .

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Problem 12

Rank the sums $\mathrm{S}_{30}$ of these series from least to greatest: $4+15+26+\ldots$ , $77+79+81+\ldots$ , $12+17+22+\ldots$ , $-178-154-130-\ldots$ .

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Problem 13

Calculate the sum: $\sum_{n=2}^{6} \frac{n-1}{n(n+1)}$ .

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Problem 14

Find the limit: $\lim _{n \rightarrow \infty}(1+x)(1+x^{2})(1+x^{4}) \cdots(1+x^{2^{n}})$ for $|x|<1$ .

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Problem 15

A spiral is made of semicircles with the first diameter $10 \mathrm{~cm}$ , each next being $\frac{4}{5}$ of the last.
1) Find the total length of the spiral.
2) How far is point $E$ (midpoint of the 6th semicircle) from point $A$ ?

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Problem 16

Überprüfe die Konvergenz der geometrischen Reihen und gib die Summen an: a) $1+2+2^{2}+\ldots$ , b) $1+\frac{1}{2}+\left(\frac{1}{2}\right)^{2}+\ldots$ , c) $1+\left(-\frac{1}{2}\right)+\left(-\frac{1}{2}\right)^{2}+\ldots$ , d) $1+(-1)+(-1)^{2}+\ldots$ , e) $3+3 \cdot \frac{1}{4}+3 \cdot\left(\frac{1}{4}\right)^{2}+\ldots$ , f) $5-5 \cdot 0,1+5 \cdot 0,1^{2}-5 \cdot 0,1^{3}+\ldots$ .

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Problem 17

Find the sums of these sequences: a. $1+2+3+\ldots+1002$ b. $1+3+5+\ldots+999$ c. $6+13+20+\ldots+706$ d. $494+489+484+\ldots+4$

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Problem 18

Find the sums of these sequences: a. $1+2+3+\ldots+999$ b. $1+3+5+\ldots+1003$ c. $4+9+14+\ldots+489$ d. $293+290+287+\ldots+2$ a. The sum is:

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Problem 19

Find the sums of these sequences using Gauss's method: a. $1+2+3+\ldots+1002$ b. $1+3+5+\ldots+999$ c. $6+13+20+\ldots+706$ d. $494+489+484+\ldots+4$

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Problem 20

Calculate the sum of the sequence from 34 to 121: $34 + 35 + 36 + \ldots + 121$ . What is the total?

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Problem 21

Find the sums of these sequences without formulas: a. $1+2+3+\ldots+69$ b. $1+3+5+\ldots+1609$ c. $5+10+15+\ldots+500$ d. $1000+995+990+\ldots+5$ a. The sum is

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Problem 22

Question 3: Calculate $1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4}$ and express the result as a mixed number.

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Problem 23

Find the next number in the series: $2, 6, 12, \ldots$

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Problem 24

Find the general term of the series: 3, 5, 7, ... which can be expressed as $a_n = 3 + 2(n-1)$ .

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Problem 25

Identify the incorrect expression for the sum $-x \sum_{r=0}^{3} \frac{x^{r+1}}{r !}$ from the options given.

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Problem 26

Calculate the sum of all whole numbers from 1 to 550, which is given by the formula $\frac{n(n+1)}{2}$ .

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Problem 27

Calculate the sum of whole numbers from 1 to 880: $\sum_{n=1}^{880} n$ .

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Problem 28

Find the sum of the series: $\sum_{n=0}^{\infty} \frac{n^{10}}{10^{n}}$ .

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Problem 29

Find the interval of convergence for the series $\sum_{n=1}^{\infty} \frac{(x-1)^{n}}{3^{n}}$ .

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Problem 30

Find the sums of these sequences without formulas: a) $1+2+3+\ldots+69$ b) $1+3+5+\ldots+2009$ c) $6+12+18+\ldots+600$ d) $1000+990+\ldots+10$

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Problem 31

Find the sums of these sequences without formulas: a. $1+2+\ldots+79$ , b. $1+3+\ldots+1609$ , c. $5+10+\ldots+500$ , d. $1000+995+\ldots+5$ .

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Problem 32

Find the sums of these sequences without formulas: a. $1+2+\ldots+89$ , b. $1+3+\ldots+2009$ , c. $6+12+\ldots+600$ , d. $1000+995+\ldots+5$ .

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Problem 33

Demostrar que la serie $\sum_{n=1}^{\infty}(\operatorname{sen} n x) / n^{2}$ converge para todo $x$ , y que $f(x)$ es continua en $[0, \pi]$ . Luego, probar que $\int_{0}^{\pi} f(x) d x=2 \sum_{n=1}^{\infty} \frac{1}{(2 n-1)^{3}}$ .

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Problem 34

Calcula la suma: $\frac{1}{2}-\frac{1}{4}+\frac{1}{8}-\frac{1}{16}=$

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Problem 35

Find the sum of squares of even numbers: $2^{2}+4^{2}+6^{2}+\cdots+100^{2}$ . Use $1^{2}+2^{2}+\cdots+n^{2}=\frac{1}{6} n(n+1)(2 n+1)$ .

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Problem 36

Calculate the sum from 33 to 106: $33 + 34 + 35 + \ldots + 105 + 106$ .

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Problem 37

Compare the sums $O = 3 + 5 + 7 + \ldots + 103$ and $E = 4 + 6 + 8 + \ldots + 104$ . Which is greater and by how much?

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Problem 38

Determine which sum is greater: $O = 3 + 5 + 7 + \ldots + 103$ or $E = 4 + 6 + 8 + \ldots + 104$ , and by how much.

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Problem 39

Find the sums of these sequences using Gauss's method (no formulas): a. $1+2+3+\ldots+39$ b. $1+3+5+\ldots+2609$ c. $3+6+9+\ldots+300$ d. $1000+990+\ldots+10$

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Problem 40

Calculate the sum of the numbers from 46 to 111: $46 + 47 + 48 + \ldots + 111$ . What is the total?

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Problem 41

Amy deposits \$1500 yearly in a 5% ordinary annuity. Find values at the end of years 1, 2, and 3.

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Problem 42

Ivan saves \$950 yearly in a 4% annuity. Find the total value at the end of years 1, 2, and 3.

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Problem 43

Find the sum: $\sum_{i=1}^{8}-2 \cdot 6^{i-1}$ .

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Problem 44

Find the sum of the first 40 terms of the arithmetic sequence where the 8th term is $23$ and the 21st term is $75$ .

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Problem 45

How many gifts did 'True Love' receive on the 12 days of Christmas? Show your work using $n$ gifts on day $n$ .

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Problem 46

Calculate the sum: $\sum_{i=1}^{8} (2i + 1)$ .

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Problem 47

Calculate the sum: $\sum_{n=1}^{12}\left(\frac{n}{2}-9\right)$ .

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Problem 48

Patricia's garbage bills are \$34.76, \$41.72, \$36.90, \$37.89, \$43.04, and \$36.65. What is her total spending?

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Problem 49

Given $\sum_{i=3}^{117} a_{i}=30$ and $\sum_{i=3}^{117} b_{i}=18$ , find: (a) $\sum_{i=3}^{117}(a_{i}+b_{i})$ and (b) $\sum_{i=3}^{117}(19 a_{i}-10 b_{i})$ .

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Problem 50

Compare the sums $P=7+9+11+\ldots+107$ and $Q=3+5+7+\ldots+105$ . Which is greater, $P$ or $Q$ , and by how much?

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Problem 51

Find the sums of these sequences using Gauss's method (no formulas): a. $1+2+3+\ldots+29$ b. $1+3+5+\ldots+2409$ c. $6+12+18+\ldots+600$ d. $1000+990+980+\ldots+10$

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Problem 52

Find the sum of the infinite geometric series with $a_{n}=64\left(\frac{1}{4}\right)^{n-1}$ .

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Problem 53

An investor deposits \$1200 yearly in an annuity at 7% interest. Find the total value after 1, 2, and 3 years.

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Problem 54

Calculate the mean of the numbers $64, 63, 65, 64, 64$ and round it to the nearest tenth.

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Problem 55

Charlotte deposits \$100 monthly into an IRA with 6% interest. How much will she have after 20 years?

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Problem 56

Find the sum of the series $\frac{25}{72}+\frac{25}{90}+\frac{25}{110}+\cdots+\frac{25}{9900}$ . Choices: (A) 2.25 (B) 2.5 (C) 2.8125 (D) 2.875 (E) $3.044 i 9$ .

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Problem 57

Calculate the sum of the series $\frac{25}{72}+\frac{25}{90}+\frac{25}{110}+\cdots+\frac{25}{9900}$ . Choose from: (A) 2.25, (B) 2.5, (C) 2.8125, (D) 2.875, (E) 3.044.

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Problem 58

श्रृंखला $\frac{25}{72}+\frac{25}{90}+\frac{25}{110}+\cdots+\frac{25}{9900}$ का योग ज्ञात करें। विकल्प: (A) 2.25 (B) 2.5 (C) 2.8125 (D) 2.875 (E) $3.044 i 9$

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Problem 59

Evaluate the geometric series: $-1 + 6 - 36 + 216 \ldots$ for $n=7$ .

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Problem 60

Calculate the sum $\sum_{k=1}^{9}(-4)^{k-1}$ .

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Problem 61

Calculate the sum of the sequence: $\frac{1}{3}+\frac{1}{9}$ .

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Problem 62

Calculate the sum of the sequence: $\frac{2}{3}+\frac{2}{9}$ .

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Problem 63

Calculate the sum of the series: $\frac{1}{3}+\frac{1}{9}+\frac{1}{27}$ .

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Problem 64

Calculate the sum of the series: $\frac{2}{3} + \frac{2}{9} + \frac{2}{27}$ .

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Problem 65

Calculate the sum: $\sum_{k=1}^{9}(0.6 k+3.4)$ .

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Problem 66

Find the sum of the first five prime numbers: 2, 3, 5, 7, and 11. What is the result?

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Problem 67

Find the sum of the first 15 terms of the geometric series: $\frac{1}{6}, \frac{1}{2}, \frac{3}{2}, \frac{9}{2}, \ldots$

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Problem 68

Find the sum of the first 15 terms of the geometric series: $\frac{1}{6}+\frac{1}{2}+\frac{3}{2}+\frac{9}{2}+\ldots$

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Problem 69

Calculate the sum of the geometric series: $\sum_{\mathrm{k}=1}^{8}(-3)(3)^{\mathrm{k}-1}$ . Options: 9840, -9000, -9840, -24.

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Problem 70

Find the sum of the first 8 terms of the geometric series: $3, 6, 12, 24, \ldots$

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Problem 71

Calculate the sum of the geometric series with $n=5$ , $r=2$ , and $a=3$ . Options: 93, 95, 105, 120.

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Problem 72

Calculate the sum of the geometric series with $n=6$ , $r=\frac{1}{4}$ , and $a=2$ .

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Problem 73

Calculate the sum of the first nine terms of the geometric series: 1, $-3$ , 9, $-27$ , ...

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Problem 74

A tennis ball dropped from 10 ft bounces half as high each time. What is the total height after 6 bounces? $26.38 \mathrm{ft}$ $27.38 \mathrm{ft}$ $28.38 \mathrm{ft}$ $29.38 \mathrm{ft}$

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Problem 75

Which option is NOT a representation of the sum of the first $n$ numbers? A. $1+2+3+\cdots+n$ B. $1+2+3+\cdots$ C. $1+2+3+\cdots+(n-1)+n$ D. $\sum^{n} i$

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Problem 76

Find the sum of the first 100 numbers using Gauss's method, which totals to $10,100$ .

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Problem 77

Use Gauss's method to find the sum of numbers from 1 to 1000. Show your work like for 1 to 8, using "..." for efficiency.

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Problem 78

The student is wrong because the sum of the first $n$ numbers is given by $\frac{n(n+1)}{2}$ , not linear.

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Problem 79

Find the sum of the first $n$ terms of the sequence where the $k$ th term is $2 + 3k$ .

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Problem 80

Find the total number of student blood donors: Type O is 40, Type A is 30, Type B is 20, Type AB is 15.

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Problem 81

Rewrite the series $1+4+9+16+25+36$ using sigma notation.

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Problem 82

Calculate the sum of the series: $\frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \frac{1}{81}$ .

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Problem 83

Find the coefficients $c_{n}$ for the heat flow problem given $u(x, t)=\sum_{n=1}^{\infty} c_{n} e^{-n^{2} t} \sin (n x)$ and $u_{4}(x, 0)=f(x)$ .

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Problem 84

Find the sum to infinity of a geometric progression with terms $8k$ , $-12$ , and $2k$ where $k$ is negative.

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Problem 85

Calculate the sum of the series: $\sum_{n=1}^{100}(2 n-9)$ . What is the result?

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Problem 86

Calculate the sum of the series: $\sum_{n=1}^{80}(2 n-5)$ . What is the result?

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Problem 87

Calculate the sum of the series: $\sum_{n=1}^{60}(3 n-8)$ .

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Problem 88

Calculate the sum of the arithmetic series: $7 + 13 + 19 + 25 + \ldots + 109$ .

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Problem 89

Calculate the sum of the series: $\sum_{n=1}^{90}(3 n-4)$ . What is the result?

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Problem 90

Find the sum of the arithmetic series: $7 + 11 + 15 + 19 + \ldots + 59$ . Simplify your answer.

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Problem 91

Is $\sum P(x)=1$ for $P(x)$ values: 0.1074, 0.3020, 0.3397, 0.1911, 0.0537, 0.0061? A. Yes B. No, $\sum P(x)=\square$ .

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Problem 92

Find $E$ where $E=\sum[x \cdot P(x)]$ for given values of $x$ and $P(x)$ , rounding to four decimal places.

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Problem 93

Calculate the sum: $\sum_{m=1}^{7}(5 m+5)$ .

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Problem 94

Calculate the sum of the first 130 natural numbers: $1 + 2 + 3 + \ldots + 130$ . What is the result?

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Problem 95

Calculate the sum of the first 78 integers: $\sum_{k=1}^{78} k=\square \text { (Provide your answer.) }$

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Problem 96

Berechnen Sie $O_{n}=\frac{1}{n} \cdot\left(\frac{1}{n}\right)^{2}+\frac{1}{n} \cdot\left(\frac{2}{n}\right)^{2}+\ldots+\frac{1}{n} \cdot\left(\frac{n}{n}\right)^{2}$ .

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Problem 97

Find the last four terms of the expansion of $\left(2 x^{3}+3 y^{2}\right)^{7}$ after these: $128 x^{21}+1,344 x^{18} y^{2}+6,048 x^{15} y^{4}+15,120 x^{12} y^{6}+\cdots$ .

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Problem 98

Javier makes trail mix with the recipe: $\frac{3}{4} + \frac{3}{4} + \frac{1}{2} + \frac{1}{2} + \frac{1}{3} + \frac{2}{3}$ . He divides it into 14 equal bags. Find the amount in each bag as a simplified fraction.

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Problem 99

Expand the function $f(x)=7 x^{2}+9 x$ in a Fourier series for $0<x<5$ . Find $c_{0}$ , $g_{1}(n, x)$ , and $g_{2}(n, x)$ .

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Problem 100

Find the sum of the first five terms of a geometric series with first term -2 and common ratio $\frac{1}{3}$ .

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

Series

Problem 1

Determine if the series converges or diverges: ∑n=1∞[(67)n−32n]\sum_{n=1}^{\infty}\left[\left(\frac{6}{7}\right)^{n}-\frac{3}{2^{n}}\right]n=1∑∞​[(76​)n−2n3​]. If it converges, find the sum.

Problem 2

Find the sum of the series ∑n=1∞1n(n+2)\sum_{n=1}^{\infty} \frac{1}{n(n+2)}∑n=1∞​n(n+2)1​.

Problem 3

Problem 4

Problem 5

Problem 6

Calculate the total of sales volumes: 2,525 million, 4,716 million, 7,241 million, and 9,096 million.

Problem 7

What is the sum of the series −2−3−4−5−⋯−101-2-3-4-5-\cdots-101−2−3−4−5−⋯−101?

Problem 8

Express the polynomial 52x3+3x4+72x5+4x6\frac{5}{2} x^{3}+3 x^{4}+\frac{7}{2} x^{5}+4 x^{6}25​x3+3x4+27​x5+4x6 in sigma notation. Which option is correct?

Problem 9

Find the sum of the series: 1+2+3+…+3041 + 2 + 3 + \ldots + 3041+2+3+…+304.

Problem 10

计算从 (13+23)\left(\frac{1}{3}+\frac{2}{3}\right)(31​+32​) 到 (1100+⋯+99100)\left(\frac{1}{100}+\cdots+\frac{99}{100}\right)(1001​+⋯+10099​) 的总和。

Problem 11

Find the sum of the first 31 terms of an arithmetic series where t1=−9t_{1}=-9t1​=−9 and t2=9t_{2}=9t2​=9.

Problem 12

Rank the sums S30\mathrm{S}_{30}S30​ of these series from least to greatest: 4+15+26+…4+15+26+\ldots4+15+26+…, 77+79+81+…77+79+81+\ldots77+79+81+…, 12+17+22+…12+17+22+\ldots12+17+22+…, −178−154−130−…-178-154-130-\ldots−178−154−130−….

Problem 13

Calculate the sum: ∑n=26n−1n(n+1)\sum_{n=2}^{6} \frac{n-1}{n(n+1)}∑n=26​n(n+1)n−1​.

Problem 14

Find the limit: lim⁡n→∞(1+x)(1+x2)(1+x4)⋯(1+x2n)\lim _{n \rightarrow \infty}(1+x)(1+x^{2})(1+x^{4}) \cdots(1+x^{2^{n}})limn→∞​(1+x)(1+x2)(1+x4)⋯(1+x2n) for ∣x∣<1|x|<1∣x∣<1.

Problem 15

A spiral is made of semicircles with the first diameter 10 cm10 \mathrm{~cm}10 cm, each next being 45\frac{4}{5}54​ of the last. 1) Find the total length of the spiral.2) How far is point EEE (midpoint of the 6th semicircle) from point AAA?

Problem 16

Problem 17

Find the sums of these sequences: a. 1+2+3+…+10021+2+3+\ldots+10021+2+3+…+1002 b. 1+3+5+…+9991+3+5+\ldots+9991+3+5+…+999 c. 6+13+20+…+7066+13+20+\ldots+7066+13+20+…+706 d. 494+489+484+…+4494+489+484+\ldots+4494+489+484+…+4

Problem 18

Find the sums of these sequences: a. 1+2+3+…+9991+2+3+\ldots+9991+2+3+…+999 b. 1+3+5+…+10031+3+5+\ldots+10031+3+5+…+1003 c. 4+9+14+…+4894+9+14+\ldots+4894+9+14+…+489 d. 293+290+287+…+2293+290+287+\ldots+2293+290+287+…+2 a. The sum is:

Problem 19

Find the sums of these sequences using Gauss's method: a. 1+2+3+…+10021+2+3+\ldots+10021+2+3+…+1002 b. 1+3+5+…+9991+3+5+\ldots+9991+3+5+…+999 c. 6+13+20+…+7066+13+20+\ldots+7066+13+20+…+706 d. 494+489+484+…+4494+489+484+\ldots+4494+489+484+…+4

Problem 20

Calculate the sum of the sequence from 34 to 121: 34+35+36+…+12134 + 35 + 36 + \ldots + 12134+35+36+…+121. What is the total?

Problem 21

Find the sums of these sequences without formulas: a. 1+2+3+…+691+2+3+\ldots+691+2+3+…+69 b. 1+3+5+…+16091+3+5+\ldots+16091+3+5+…+1609 c. 5+10+15+…+5005+10+15+\ldots+5005+10+15+…+500 d. 1000+995+990+…+51000+995+990+\ldots+51000+995+990+…+5 a. The sum is

Problem 22

Question 3: Calculate 1+12+13+141 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4}1+21​+31​+41​ and express the result as a mixed number.

Problem 23

Find the next number in the series: 2,6,12,…2, 6, 12, \ldots2,6,12,…

Problem 24

Find the general term of the series: 3, 5, 7, ... which can be expressed as an=3+2(n−1)a_n = 3 + 2(n-1)an​=3+2(n−1).

Problem 25

Identify the incorrect expression for the sum −x∑r=03xr+1r!-x \sum_{r=0}^{3} \frac{x^{r+1}}{r !}−xr=0∑3​r!xr+1​ from the options given.

Problem 26

Calculate the sum of all whole numbers from 1 to 550, which is given by the formula n(n+1)2\frac{n(n+1)}{2}2n(n+1)​.

Problem 27

Calculate the sum of whole numbers from 1 to 880: ∑n=1880n\sum_{n=1}^{880} nn=1∑880​n.

Problem 28

Find the sum of the series: ∑n=0∞n1010n\sum_{n=0}^{\infty} \frac{n^{10}}{10^{n}}∑n=0∞​10nn10​.

Problem 29

Find the interval of convergence for the series ∑n=1∞(x−1)n3n\sum_{n=1}^{\infty} \frac{(x-1)^{n}}{3^{n}}∑n=1∞​3n(x−1)n​.

Problem 30

Find the sums of these sequences without formulas: a) 1+2+3+…+691+2+3+\ldots+691+2+3+…+69 b) 1+3+5+…+20091+3+5+\ldots+20091+3+5+…+2009 c) 6+12+18+…+6006+12+18+\ldots+6006+12+18+…+600 d) 1000+990+…+101000+990+\ldots+101000+990+…+10

Problem 31

Find the sums of these sequences without formulas: a. 1+2+…+791+2+\ldots+791+2+…+79, b. 1+3+…+16091+3+\ldots+16091+3+…+1609, c. 5+10+…+5005+10+\ldots+5005+10+…+500, d. 1000+995+…+51000+995+\ldots+51000+995+…+5.

Problem 32

Find the sums of these sequences without formulas: a. 1+2+…+891+2+\ldots+891+2+…+89, b. 1+3+…+20091+3+\ldots+20091+3+…+2009, c. 6+12+…+6006+12+\ldots+6006+12+…+600, d. 1000+995+…+51000+995+\ldots+51000+995+…+5.

Problem 33

Problem 34

Calcula la suma: 12−14+18−116=\frac{1}{2}-\frac{1}{4}+\frac{1}{8}-\frac{1}{16}=21​−41​+81​−161​=

Problem 35

Find the sum of squares of even numbers: 22+42+62+⋯+10022^{2}+4^{2}+6^{2}+\cdots+100^{2}22+42+62+⋯+1002. Use 12+22+⋯+n2=16n(n+1)(2n+1)1^{2}+2^{2}+\cdots+n^{2}=\frac{1}{6} n(n+1)(2 n+1)12+22+⋯+n2=61​n(n+1)(2n+1).

Problem 36

Calculate the sum from 33 to 106: 33+34+35+…+105+10633 + 34 + 35 + \ldots + 105 + 10633+34+35+…+105+106.

Problem 37

Compare the sums O=3+5+7+…+103O = 3 + 5 + 7 + \ldots + 103O=3+5+7+…+103 and E=4+6+8+…+104E = 4 + 6 + 8 + \ldots + 104E=4+6+8+…+104. Which is greater and by how much?

Problem 38

Determine which sum is greater: O=3+5+7+…+103O = 3 + 5 + 7 + \ldots + 103O=3+5+7+…+103 or E=4+6+8+…+104E = 4 + 6 + 8 + \ldots + 104E=4+6+8+…+104, and by how much.

Problem 39

Find the sums of these sequences using Gauss's method (no formulas): a. 1+2+3+…+391+2+3+\ldots+391+2+3+…+39 b. 1+3+5+…+26091+3+5+\ldots+26091+3+5+…+2609 c. 3+6+9+…+3003+6+9+\ldots+3003+6+9+…+300 d. 1000+990+…+101000+990+\ldots+101000+990+…+10

Problem 40

Calculate the sum of the numbers from 46 to 111: 46+47+48+…+11146 + 47 + 48 + \ldots + 11146+47+48+…+111. What is the total?

Problem 41

Amy deposits \$1500 yearly in a 5% ordinary annuity. Find values at the end of years 1, 2, and 3.

Problem 42

Ivan saves \$950 yearly in a 4% annuity. Find the total value at the end of years 1, 2, and 3.

Determine if the series converges or diverges: $\sum_{n=1}^{\infty}\left[\left(\frac{6}{7}\right)^{n}-\frac{3}{2^{n}}\right]$ . If it converges, find the sum.

Find the sum of the series $\sum_{n=1}^{\infty} \frac{1}{n(n+2)}$ .

What is the sum of the series $-2-3-4-5-\cdots-101$ ?

Express the polynomial $\frac{5}{2} x^{3}+3 x^{4}+\frac{7}{2} x^{5}+4 x^{6}$ in sigma notation. Which option is correct?

Find the sum of the series: $1 + 2 + 3 + \ldots + 304$ .

计算从 $\left(\frac{1}{3}+\frac{2}{3}\right)$ 到 $\left(\frac{1}{100}+\cdots+\frac{99}{100}\right)$ 的总和。

Find the sum of the first 31 terms of an arithmetic series where $t_{1}=-9$ and $t_{2}=9$ .

Rank the sums $\mathrm{S}_{30}$ of these series from least to greatest: $4+15+26+\ldots$ , $77+79+81+\ldots$ , $12+17+22+\ldots$ , $-178-154-130-\ldots$ .

Calculate the sum: $\sum_{n=2}^{6} \frac{n-1}{n(n+1)}$ .

Find the limit: $\lim _{n \rightarrow \infty}(1+x)(1+x^{2})(1+x^{4}) \cdots(1+x^{2^{n}})$ for $|x|<1$ .

A spiral is made of semicircles with the first diameter $10 \mathrm{~cm}$ , each next being $\frac{4}{5}$ of the last.
1) Find the total length of the spiral.
2) How far is point $E$ (midpoint of the 6th semicircle) from point $A$ ?

Find the sums of these sequences: a. $1+2+3+\ldots+1002$ b. $1+3+5+\ldots+999$ c. $6+13+20+\ldots+706$ d. $494+489+484+\ldots+4$

Find the sums of these sequences: a. $1+2+3+\ldots+999$ b. $1+3+5+\ldots+1003$ c. $4+9+14+\ldots+489$ d. $293+290+287+\ldots+2$ a. The sum is:

Find the sums of these sequences using Gauss's method: a. $1+2+3+\ldots+1002$ b. $1+3+5+\ldots+999$ c. $6+13+20+\ldots+706$ d. $494+489+484+\ldots+4$

Calculate the sum of the sequence from 34 to 121: $34 + 35 + 36 + \ldots + 121$ . What is the total?

Find the sums of these sequences without formulas: a. $1+2+3+\ldots+69$ b. $1+3+5+\ldots+1609$ c. $5+10+15+\ldots+500$ d. $1000+995+990+\ldots+5$ a. The sum is

Question 3: Calculate $1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4}$ and express the result as a mixed number.

Find the next number in the series: $2, 6, 12, \ldots$

Find the general term of the series: 3, 5, 7, ... which can be expressed as $a_n = 3 + 2(n-1)$ .

Identify the incorrect expression for the sum $-x \sum_{r=0}^{3} \frac{x^{r+1}}{r !}$ from the options given.

Calculate the sum of all whole numbers from 1 to 550, which is given by the formula $\frac{n(n+1)}{2}$ .

Calculate the sum of whole numbers from 1 to 880: $\sum_{n=1}^{880} n$ .

Find the sum of the series: $\sum_{n=0}^{\infty} \frac{n^{10}}{10^{n}}$ .

Find the interval of convergence for the series $\sum_{n=1}^{\infty} \frac{(x-1)^{n}}{3^{n}}$ .

Find the sums of these sequences without formulas: a) $1+2+3+\ldots+69$ b) $1+3+5+\ldots+2009$ c) $6+12+18+\ldots+600$ d) $1000+990+\ldots+10$

Find the sums of these sequences without formulas: a. $1+2+\ldots+79$ , b. $1+3+\ldots+1609$ , c. $5+10+\ldots+500$ , d. $1000+995+\ldots+5$ .

Find the sums of these sequences without formulas: a. $1+2+\ldots+89$ , b. $1+3+\ldots+2009$ , c. $6+12+\ldots+600$ , d. $1000+995+\ldots+5$ .

Calcula la suma: $\frac{1}{2}-\frac{1}{4}+\frac{1}{8}-\frac{1}{16}=$

Find the sum of squares of even numbers: $2^{2}+4^{2}+6^{2}+\cdots+100^{2}$ . Use $1^{2}+2^{2}+\cdots+n^{2}=\frac{1}{6} n(n+1)(2 n+1)$ .

Calculate the sum from 33 to 106: $33 + 34 + 35 + \ldots + 105 + 106$ .

Compare the sums $O = 3 + 5 + 7 + \ldots + 103$ and $E = 4 + 6 + 8 + \ldots + 104$ . Which is greater and by how much?

Determine which sum is greater: $O = 3 + 5 + 7 + \ldots + 103$ or $E = 4 + 6 + 8 + \ldots + 104$ , and by how much.

Find the sums of these sequences using Gauss's method (no formulas): a. $1+2+3+\ldots+39$ b. $1+3+5+\ldots+2609$ c. $3+6+9+\ldots+300$ d. $1000+990+\ldots+10$

Calculate the sum of the numbers from 46 to 111: $46 + 47 + 48 + \ldots + 111$ . What is the total?

Find the sum: $\sum_{i=1}^{8}-2 \cdot 6^{i-1}$ .

Find the sum of the first 40 terms of the arithmetic sequence where the 8th term is $23$ and the 21st term is $75$ .

How many gifts did 'True Love' receive on the 12 days of Christmas? Show your work using $n$ gifts on day $n$ .

Calculate the sum: $\sum_{i=1}^{8} (2i + 1)$ .

Calculate the sum: $\sum_{n=1}^{12}\left(\frac{n}{2}-9\right)$ .

Compare the sums $P=7+9+11+\ldots+107$ and $Q=3+5+7+\ldots+105$ . Which is greater, $P$ or $Q$ , and by how much?

Find the sums of these sequences using Gauss's method (no formulas): a. $1+2+3+\ldots+29$ b. $1+3+5+\ldots+2409$ c. $6+12+18+\ldots+600$ d. $1000+990+980+\ldots+10$

Find the sum of the infinite geometric series with $a_{n}=64\left(\frac{1}{4}\right)^{n-1}$ .

Calculate the mean of the numbers $64, 63, 65, 64, 64$ and round it to the nearest tenth.

Find the sum of the series $\frac{25}{72}+\frac{25}{90}+\frac{25}{110}+\cdots+\frac{25}{9900}$ . Choices: (A) 2.25 (B) 2.5 (C) 2.8125 (D) 2.875 (E) $3.044 i 9$ .

Calculate the sum of the series $\frac{25}{72}+\frac{25}{90}+\frac{25}{110}+\cdots+\frac{25}{9900}$ . Choose from: (A) 2.25, (B) 2.5, (C) 2.8125, (D) 2.875, (E) 3.044.

श्रृंखला $\frac{25}{72}+\frac{25}{90}+\frac{25}{110}+\cdots+\frac{25}{9900}$ का योग ज्ञात करें। विकल्प: (A) 2.25 (B) 2.5 (C) 2.8125 (D) 2.875 (E) $3.044 i 9$

Evaluate the geometric series: $-1 + 6 - 36 + 216 \ldots$ for $n=7$ .

Calculate the sum $\sum_{k=1}^{9}(-4)^{k-1}$ .

Calculate the sum of the sequence: $\frac{1}{3}+\frac{1}{9}$ .

Calculate the sum of the sequence: $\frac{2}{3}+\frac{2}{9}$ .

Calculate the sum of the series: $\frac{1}{3}+\frac{1}{9}+\frac{1}{27}$ .

Calculate the sum of the series: $\frac{2}{3} + \frac{2}{9} + \frac{2}{27}$ .

Calculate the sum: $\sum_{k=1}^{9}(0.6 k+3.4)$ .

Find the sum of the first 15 terms of the geometric series: $\frac{1}{6}, \frac{1}{2}, \frac{3}{2}, \frac{9}{2}, \ldots$

Find the sum of the first 15 terms of the geometric series: $\frac{1}{6}+\frac{1}{2}+\frac{3}{2}+\frac{9}{2}+\ldots$

Calculate the sum of the geometric series: $\sum_{\mathrm{k}=1}^{8}(-3)(3)^{\mathrm{k}-1}$ . Options: 9840, -9000, -9840, -24.

Find the sum of the first 8 terms of the geometric series: $3, 6, 12, 24, \ldots$

Calculate the sum of the geometric series with $n=5$ , $r=2$ , and $a=3$ . Options: 93, 95, 105, 120.

Calculate the sum of the geometric series with $n=6$ , $r=\frac{1}{4}$ , and $a=2$ .

Calculate the sum of the first nine terms of the geometric series: 1, $-3$ , 9, $-27$ , ...

A tennis ball dropped from 10 ft bounces half as high each time. What is the total height after 6 bounces? $26.38 \mathrm{ft}$ $27.38 \mathrm{ft}$ $28.38 \mathrm{ft}$ $29.38 \mathrm{ft}$