Solve for X help

Question

Solve for x:

∞

∑ 4x⁵ⁿ = 40

n = 1

x = ?

Answer 1

Let's write out the series first:

4x⁵ + 4x¹⁰ + 4x¹⁵ + 4x²⁰ + ... = 40

Then divide both sides by 4, we have:

x⁵ + x¹⁰ + x¹⁵ + x²⁰ + ... = 10

This is a geometric series with a common ratio of x⁵.

The sum of an infinite geometric series is (a₁) / (1 - r) where a₁ is the first term and r is the common ratio.

Let's use this formula, then solve for x:

x⁵ / (1 - x⁵) = 10

x⁵ = 10 - 10x⁵

11x⁵ = 10

x⁵ = 10/11

x = (10/11)^1/5 ≈ .9811

Let me know if you have any questions!