Chapter 12.3, Problem 43AYU

To determine

To find: The sum of series.

Answer to Problem 43AYU

The sum of series is $S_n=\left(1-2^n\right)$ .

Explanation of Solution

Given information:

The series is $-1-2-4-\mathrm{8............}-\left(2^{n-1}\right)$ .

Calculation:

For the given series $a_1=-1$ , $a_2=-2$ and $a_3=-4$ .

Calculate $\frac{a_2}{a_1}$ .

  $\begin{array}{c}\frac{a_2}{a_1}=\frac{-2}{-1}\\ =2\end{array}$

Calculate $\frac{a_3}{a_1}$ .

  $\begin{array}{c}\frac{a_3}{a_1}=\frac{-4}{-2}\\ =2\end{array}$

Since the ratio of successive terms is same it is a geometric series with common ratio $r=2$ .

The sum of first $n$ terms of a geometric sequence is calculated by,

  $S_n=a_1\frac{1-r^n}{1-r}$

Substitute the given values.

  $\begin{array}{l}{S}_n=-1\left(\frac{1-2^n}{1-2}\right)\\ S_n=-1\frac{\left(1-2^n\right)}{-1}\\ S_n=\left(1-2^n\right)\end{array}$

Therefore, the sum of series is $S_n=\left(1-2^n\right)$ .