Chapter 8, Problem 98RE

To determine

To find: Whether given statement is True or False.

Answer to Problem 98RE

TRUE

Explanation of Solution

Given information:

  ${\displaystyle \sum _{j=1}^6{3}^j}={\displaystyle \sum _{j=3}^8{3}^{j-2}}$

  ${\displaystyle \sum _{j=1}^6{3}^j}={\displaystyle \sum _{j=3}^8{3}^{j-2}}$

Taking Left hand side

We get

Left hand side $={\displaystyle \sum _{j=1}^6{3}^j}$

On expanding Sigma

We get

  $\Rightarrow$ Left hand side $=3^1+3^2+3^3+3^4+3^5+3^6$

Taking Right hand side

We get

Right hand side $={\displaystyle \sum _{j=3}^8{3}^{j-2}}$

On expanding Sigma

We get

  $\Rightarrow$ Right hand side $=3^{3-2}+3^{4-2}+3^{5-2}+3^{6-2}+3^{7-2}+3^{8-2}$

  $\Rightarrow$ Right hand side $=3^1+3^2+3^3+3^4+3^5+3^6$

  $\Rightarrow$ Right hand side $=$ Left hand side

Therefore,

Given statement

  ${\displaystyle \sum _{j=1}^6{3}^j}={\displaystyle \sum _{j=3}^8{3}^{j-2}}$

Is TRUE.