Q. 15 - Calculus

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.

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

Step-by-Step Solution

Answer

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

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.

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

Write the terms in sigma notation.

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