Chapter 11.5, Problem 26WE

a.

To determine

To calculate:

Sum of the first ten powers of $5$ .

Answer to Problem 26WE

Sum of the first ten powers of $5$ is $12207030$ .

Explanation of Solution

Give information:

The first ten powers of $5$ .

Calculation:

Given is The first ten powers of $5$ ,

The sum is $5^1+5^2+\mathrm{...}+5^{10}$

This is geometric series,

Using the formula for the sum of geometric series is,

  $\frac{t_1\left(1-r^n\right)}{1-r}\left(r\ne 1\right)$

so the sum is,

  $\begin{array}{l}\frac{5\left(1-5^{10}\right)}{1-5}\\ =5\left(5^{10}-1\right)/4\\ =12207030\end{array}$

Therefore, Sum of the first ten powers of $5$ is $12207030$ .

b.

To determine

To calculate:

Sum of the first ten powers of $5$ that are perfect square.

Answer to Problem 26WE

Sum of the first ten powers of $5$ that are perfect square is $99341074625650$ .

Explanation of Solution

Give information:

The first ten powers of $5$ that are perfect square.

Calculation:

Given is The first ten powers of $5$ ,

The sum is $5^2+5^4+\mathrm{...}+5^{20}$

This is geometric series,

Using the formula for the sum of geometric series is,

  $\frac{t_1\left(1-r^n\right)}{1-r}\left(r\ne 1\right)$

So the sum is,

  $\begin{array}{l}\frac{25\left(1-5^{20}\right)}{1-25}\\ =25\left(5^{20}-1\right)/24\\ =99341074625650\end{array}$

Therefore, Sum of the first ten powers of $5$ that are perfect square is $99341074625650$ .