Q. 39 - Calculus

Question

Evaluate the finite sums.

$\sum _{k=0}^{1000} 2^k$.

Step-by-Step Solution

Answer

The sum of the series $\sum _{k=0}^{1000} 2^k$ is, $2^{1001}-1$.

Step 1 . Given Information

$\sum _{k=0}^{1000} 2^k$.

Step 2 . Let us expand the given series.

$\sum _{k=0}^{1000} 2^k=1+2+4+...+2^{1000}$

The first term in the series $a=1$.

The common ratio $r=2$ with the number of terms $n=1001$.

Step 3 . The finite sum of the geometric series with ratio greater than 1 is given by,

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

Substitute the known values in the formula.

$$\begin{gathered} S_{1001}=\frac{1\left(2^{1001}-1\right)}{2-1} \\ =\frac{2^{1001}-1}{1} \\ =2^{1001}-1 \end{gathered}$$