Solved on Dec 14, 2023

Find the value of $x$ that satisfies the infinite series equation $\sum_{n=1}^{\infty} 2 x^{5 n} = 20$ .

STEP 1

Assumptions
1. The given series is a geometric series.
2. The series converges since it is equal to a finite number (20).
3. The formula for the sum of an infinite geometric series is $\frac{a}{1 - r}$ , where $a$ is the first term and $r$ is the common ratio.

STEP 2

Identify the first term ( $a$ ) of the geometric series. The first term occurs when $n=1$ .
$a = 2x^{5}$

STEP 3

Identify the common ratio ( $r$ ) of the geometric series. The ratio between consecutive terms is $x^5$ .
$r = x^{5}$

STEP 4

Use the formula for the sum of an infinite geometric series to set up the equation.
$\sum_{n=1}^{\infty} 2 x^{5 n} = \frac{a}{1 - r}$

STEP 5

Substitute the values of $a$ and $r$ into the equation.
$20 = \frac{2x^{5}}{1 - x^{5}}$

STEP 6

Multiply both sides of the equation by $(1 - x^{5})$ to eliminate the denominator.
$20(1 - x^{5}) = 2x^{5}$

STEP 7

Distribute the 20 on the left side of the equation.
$20 - 20x^{5} = 2x^{5}$

STEP 8

Add $20x^{5}$ to both sides of the equation to bring all the $x$ terms to one side.
$20 = 22x^{5}$

STEP 9

Divide both sides of the equation by 22 to solve for $x^{5}$ .
$x^{5} = \frac{20}{22}$

STEP 10

Simplify the fraction on the right side of the equation.
$x^{5} = \frac{10}{11}$

STEP 11

Take the fifth root of both sides to solve for $x$ .
$x = \sqrt[5]{\frac{10}{11}}$

STEP 12

Calculate the fifth root of $\frac{10}{11}$ to find the value of $x$ .
$x = \left(\frac{10}{11}\right)^{\frac{1}{5}}$
This is the value of $x$ that satisfies the given equation.

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Find the value of $x$ that satisfies the infinite series equation $\sum_{n=1}^{\infty} 2 x^{5 n} = 20$ .

STEP 1

Assumptions
1. The given series is a geometric series.
2. The series converges since it is equal to a finite number (20).
3. The formula for the sum of an infinite geometric series is $\frac{a}{1 - r}$ , where $a$ is the first term and $r$ is the common ratio.

STEP 2

Identify the first term ( $a$ ) of the geometric series. The first term occurs when $n=1$ .
$a = 2x^{5}$

STEP 3

Identify the common ratio ( $r$ ) of the geometric series. The ratio between consecutive terms is $x^5$ .
$r = x^{5}$

STEP 4

Use the formula for the sum of an infinite geometric series to set up the equation.
$\sum_{n=1}^{\infty} 2 x^{5 n} = \frac{a}{1 - r}$

STEP 5

Substitute the values of $a$ and $r$ into the equation.
$20 = \frac{2x^{5}}{1 - x^{5}}$

STEP 6

Multiply both sides of the equation by $(1 - x^{5})$ to eliminate the denominator.
$20(1 - x^{5}) = 2x^{5}$

STEP 7

Distribute the 20 on the left side of the equation.
$20 - 20x^{5} = 2x^{5}$

STEP 8

Add $20x^{5}$ to both sides of the equation to bring all the $x$ terms to one side.
$20 = 22x^{5}$

STEP 9

Divide both sides of the equation by 22 to solve for $x^{5}$ .
$x^{5} = \frac{20}{22}$

STEP 10

Simplify the fraction on the right side of the equation.
$x^{5} = \frac{10}{11}$

STEP 11

Take the fifth root of both sides to solve for $x$ .
$x = \sqrt[5]{\frac{10}{11}}$

STEP 12

Calculate the fifth root of $\frac{10}{11}$ to find the value of $x$ .
$x = \left(\frac{10}{11}\right)^{\frac{1}{5}}$
This is the value of $x$ that satisfies the given equation.

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Find the value of xxx that satisfies the infinite series equation ∑n=1∞2x5n=20\sum_{n=1}^{\infty} 2 x^{5 n} = 20∑n=1∞​2x5n=20.

STEP 1

Assumptions1. The given series is a geometric series.2. The series converges since it is equal to a finite number (20).3. The formula for the sum of an infinite geometric series is a1−r\frac{a}{1 - r}1−ra​, where aaa is the first term and rrr is the common ratio.

STEP 2

Identify the first term (aaa) of the geometric series. The first term occurs when n=1n=1n=1.a=2x5a = 2x^{5}a=2x5

STEP 3

Identify the common ratio (rrr) of the geometric series. The ratio between consecutive terms is x5x^5x5.r=x5r = x^{5}r=x5

STEP 4

Use the formula for the sum of an infinite geometric series to set up the equation.∑n=1∞2x5n=a1−r\sum_{n=1}^{\infty} 2 x^{5 n} = \frac{a}{1 - r}n=1∑∞​2x5n=1−ra​

STEP 5

Substitute the values of aaa and rrr into the equation.20=2x51−x520 = \frac{2x^{5}}{1 - x^{5}}20=1−x52x5​

STEP 6

Multiply both sides of the equation by (1−x5)(1 - x^{5})(1−x5) to eliminate the denominator.20(1−x5)=2x520(1 - x^{5}) = 2x^{5}20(1−x5)=2x5

STEP 7

Distribute the 20 on the left side of the equation.20−20x5=2x520 - 20x^{5} = 2x^{5}20−20x5=2x5

STEP 8

Add 20x520x^{5}20x5 to both sides of the equation to bring all the xxx terms to one side.20=22x520 = 22x^{5}20=22x5

STEP 9

Divide both sides of the equation by 22 to solve for x5x^{5}x5.x5=2022x^{5} = \frac{20}{22}x5=2220​

STEP 10

Simplify the fraction on the right side of the equation.x5=1011x^{5} = \frac{10}{11}x5=1110​

STEP 11

Take the fifth root of both sides to solve for xxx.x=10115x = \sqrt[5]{\frac{10}{11}}x=51110​​

STEP 12

Calculate the fifth root of 1011\frac{10}{11}1110​ to find the value of xxx.x=(1011)15x = \left(\frac{10}{11}\right)^{\frac{1}{5}}x=(1110​)51​This is the value of xxx that satisfies the given equation.

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Find the value of $x$ that satisfies the infinite series equation $\sum_{n=1}^{\infty} 2 x^{5 n} = 20$ .

Assumptions
1. The given series is a geometric series.
2. The series converges since it is equal to a finite number (20).
3. The formula for the sum of an infinite geometric series is $\frac{a}{1 - r}$ , where $a$ is the first term and $r$ is the common ratio.

Identify the first term ( $a$ ) of the geometric series. The first term occurs when $n=1$ .
$a = 2x^{5}$

Identify the common ratio ( $r$ ) of the geometric series. The ratio between consecutive terms is $x^5$ .
$r = x^{5}$

Use the formula for the sum of an infinite geometric series to set up the equation.
$\sum_{n=1}^{\infty} 2 x^{5 n} = \frac{a}{1 - r}$

Substitute the values of $a$ and $r$ into the equation.
$20 = \frac{2x^{5}}{1 - x^{5}}$

Multiply both sides of the equation by $(1 - x^{5})$ to eliminate the denominator.
$20(1 - x^{5}) = 2x^{5}$

Distribute the 20 on the left side of the equation.
$20 - 20x^{5} = 2x^{5}$

Add $20x^{5}$ to both sides of the equation to bring all the $x$ terms to one side.
$20 = 22x^{5}$

Divide both sides of the equation by 22 to solve for $x^{5}$ .
$x^{5} = \frac{20}{22}$

Simplify the fraction on the right side of the equation.
$x^{5} = \frac{10}{11}$

Take the fifth root of both sides to solve for $x$ .
$x = \sqrt[5]{\frac{10}{11}}$

Calculate the fifth root of $\frac{10}{11}$ to find the value of $x$ .
$x = \left(\frac{10}{11}\right)^{\frac{1}{5}}$
This is the value of $x$ that satisfies the given equation.