Question

Evaluating a Sum Find the sum.

$\sum _{k=1}^5{2}^{k-1}$

Short answer

The sum is 31.

Step-by-step solution

Step 1.  Given

The given sequence is$\sum _{k=1}^5{2}^{k-1}.$

Step 2.  To determine

We have to find thesum of the given sequence.

Step 3.  Calculation

We’ll plug the values of k and get rid of the summation sign to simplify:

$\begin{array}{l}\sum _{k=1}^5{2}^{k-1}\\ =2^{1-1}+2^{2-1}+2^{3-1}+2^{4-1}+2^{5-1}\\ =2^0+2^1+2^2+2^3+2^4\\ =1+2+4+8+16\\ =31\end{array}$

Hence, the sum is 31.