Solved on Dec 14, 2023

Find the value of xx that satisfies the infinite series equation n=12x5n=20\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 a1r\frac{a}{1 - r}, where aa is the first term and rr is the common ratio.

STEP 2

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

STEP 3

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

STEP 4

Use the formula for the sum of an infinite geometric series to set up the equation.
n=12x5n=a1r\sum_{n=1}^{\infty} 2 x^{5 n} = \frac{a}{1 - r}

STEP 5

Substitute the values of aa and rr into the equation.
20=2x51x520 = \frac{2x^{5}}{1 - x^{5}}

STEP 6

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

STEP 7

Distribute the 20 on the left side of the equation.
2020x5=2x520 - 20x^{5} = 2x^{5}

STEP 8

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

STEP 9

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

STEP 10

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

STEP 11

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

STEP 12

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

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