Question

Find the sum or quantity without completely expanding or calculating any sums.

Given $\sum _{k=1}^4{a}_k=7$and$,\sum _{k=0}^4{b}_k=10$,$a_0=2$. Find the value of$\sum _{k=0}^4\left(2a_k+3b_k\right)$.

Short answer

The sum is$48$.

Step-by-step solution

Step 1. Given information

$$\begin{gathered} \sum _{k=1}^4{a}_k=7 \\ \sum _{k=0}^4{b}_k=10 \\ {a}_0=2 \end{gathered}$$

Step 2. Find the sum:

$$\begin{gathered} \sum _{k=0}^4(2a_k+3b_k) \\ =\sum _{k=0}^{4}2a_k+\sum _{k=0}^{4}3b_k \\ =2a_0+\sum _{k=1}^{4}2a_k+\sum _{k=0}^{4}3b_k \\ =2a_0+2\sum _{k=1}^4{a}_k+3\sum _{k=0}^4{b}_k \\ =2\times 2+2\times 7+3\times 10 \\ =4+14+30 \\ =48 \end{gathered}$$