Chapter 11.5, Problem 25WE

a.

To determine

To calculate: The sum of twenty multiple of 2.

Answer to Problem 25WE

The sum of first twenty multiple of 2 is 420.

Explanation of Solution

Given information: The term start with 2

According to the question, $2+4+6+8+\mathrm{..............}+20\times 2$

Let, first term, $a=2$

Second term, $a_1=4$

Common difference, $d=a_1-a$

  $\Rightarrow d=4-2=2$

Calculation: $S_n=\frac{n}{2}\left[2a+\left(n-1\right)d\right]$

  $\begin{array}{l}\Rightarrow S_{20}=\frac{20}{2}\left[2\times 2+\left(20-1\right)2\right]\\ =10\left[4+19\times 2\right]\\ =10\left[4+38\right]\\ =10\times 42\\ =420\end{array}$

Hence, the sum of first twenty multiple of 2 is 420.

b.

To determine

To calculate: The sum of twenty powers of 2.

Answer to Problem 25WE

The sum of first twenty powers of 2 is 2097150.

Explanation of Solution

Given information: The term start with $2^1$

According to the question, $2^1+2^2+2^3+\mathrm{..............}+2^{20}$

Let, first term, $a=2^1$

Second term, $a_1=2^2$

Common ratio, $r=\frac{a_1}{a}$

  $\Rightarrow r=\frac{2^2}{2^1}=\frac{4}{2}=2$

Calculation: $S_n=\frac{a\left(r^n-1\right)}{r-1}$

  $\begin{array}{l}\Rightarrow S_{20}=\frac{2^1\left(2^{20}-1\right)}{2-1}\\ \Rightarrow S_{20}=\frac{2\left(1048576-1\right)}{1}\\ \Rightarrow S_{20}=2\times 1048575\\ \Rightarrow S_{20}=2097150\end{array}$

Hence, the sum of first twenty powers of 2 is 2097150.