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.

$\sum _{k=1}^{n}c{a}_k=$______

Short answer

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

Step-by-step solution

Step 1. Given information

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

Step 2. First translate the sigma notation to its expanded form and then apply the distributive property:

$\sum _{k=1}^{n}c{a}_k=c{a}_1+c{a}_2+c{a}_3+\cdots +c{a}_{n-1}+c{a}_n$

Now factor out the common terms.

$c{a}_1+c{a}_2+c{a}_3+\cdots +c{a}_{n-1}+c{a}_n=c\left(a_1+a_2+a_3+\cdots +a_{n-1}+a_n\right)$.

Now write in sigma notation.

role="math" localid="1648821440325" $c\left(a_1+a_2+a_3+\cdots +a_{n-1}+a_n\right)=c\sum _{k=1}^n{a}_k$.