Chapter 8.1, Problem 125E

To determine

Whether the given statement is true or false.

Answer to Problem 125E

The statement is true.

Explanation of Solution

Given information:

A statement is given as

  ${\displaystyle \sum _{i=1}^4\left(i^2+2i\right)}={\displaystyle \sum _{i=1}^4{i}^2}+2{\displaystyle \sum _{i=1}^{4}i}$

Formula used:

Property of sigma notation

  ${\displaystyle \sum _{k=1}^n\left(c{a}_k+d{b}_k\right)}=c{\displaystyle \sum _{k=1}^n{a}_k}+d{\displaystyle \sum _{k=1}^n{a}_k}$

Calculation:

Consider the given statement.

  ${\displaystyle \sum _{i=1}^4\left(i^2+2i\right)}={\displaystyle \sum _{i=1}^4{i}^2}+2{\displaystyle \sum _{i=1}^{4}i}$

Now, compare the above statement with ${\displaystyle \sum _{k=1}^n\left(c{a}_k+d{b}_k\right)}=c{\displaystyle \sum _{k=1}^n{a}_k}+d{\displaystyle \sum _{k=1}^n{a}_k}$ .

  $a_k=i^2,n=4,d=2,b_k=i$

Therefore, the statement ${\displaystyle \sum _{i=1}^4\left(i^2+2i\right)}={\displaystyle \sum _{i=1}^4{i}^2}+2{\displaystyle \sum _{i=1}^{4}i}$ is true.