Question

The algebra of sums: Fill in the blanks to complete the sum rules that follow. You may assume that $a_k$and $b_k$are functions defined for nonnegative integers $k$and that $c$is any real number.

$\sum _{k=1}^n\left(a_k+b_k\right)=$______

Short answer

$\sum _{k=1}^n\left(a_k+b_k\right)=\sum _{k=1}^n{a}_k+\sum _{k=1}^n{b}_k$.

Step-by-step solution

Step 1. Given information

$\sum _{k=1}^n\left(a_k+b_k\right)$.

Step 2. Write the sigma notation in expanded form.

$\sum _{k=1}^n\left(a_k+b_k\right)=\left(a_1+b_1\right)+\left(a_2+b_2\right)+\left(a_3+b_3\right)+\cdots \left(a_{n-1}+b_{n-1}\right)+\left(a_n+b_n\right)$

Now regroup the terms in sum.

role="math" localid="1648822752316" $=\left(a_1+a_2+a_3+\cdots +a_{n-1}+a_n\right)+\left(b_1+b_2+b_3+\cdots +b_{n-1}+b_n\right)$

Write the terms in sigma notation.

$=\sum _{k=1}^n{a}_k+\sum _{k=1}^n{b}_k$.