Question

Given that$a_0=-3,a_1=5,a_2=-4,a_3=2$and$\underset{k=2}{\overset{\infty }{\sum a_k}}=7$, find the value of$\underset{k=4}{\overset{\infty }{\sum a_k}}$.

Short answer

The value of$\underset{k=4}{\overset{\infty }{\sum a_k}}$is 9.

Step-by-step solution

Step 1. Given information.

Given that$a_0=-3,a_1=5,a_2=-4,a_3=2$and$\underset{k=2}{\overset{\infty }{\sum a_k}}=7$.

Step 2. Express the series in terms of known values.

Value of $\underset{k=4}{\overset{\infty }{\sum a_k}}$can be obtained as follows.

$\begin{array}{rcl}\underset{k=4}{\overset{\infty }{\sum a_k}}& =& \underset{k=2}{\overset{\infty }{\sum a_k}}-\left(a_2+a_3\right)\\ & =& 7-\left(-4+2\right)\\ & =& 7+2\\ & =& 9\end{array}$

Therefore, value of $\sum _{k=4}^{\infty }{a}_k$is 9.