Chapter 9, Problem 14CLT

To determine

To find: ${\displaystyle \sum _{n=0}^{10}9{\left(\frac{3}{4}\right)}^n}$ .

Answer to Problem 14CLT

  ${\displaystyle \sum _{n=0}^{10}9{\left(\frac{3}{4}\right)}^n}$ = 34.48

Explanation of Solution

Given information:

  ${\displaystyle \sum _{n=0}^{10}9{\left(\frac{3}{4}\right)}^n}$

  ${\displaystyle \sum _{n=0}^{10}9{\left(\frac{3}{4}\right)}^n}$

Use the Sum of series

  ${\displaystyle \sum _{i=0}^n{a}_i}=a_0+a_1+a_2+\mathrm{......}+a_n$

Therefore

  ${\displaystyle \sum _{n=0}^{10}9{\left(\frac{3}{4}\right)}^n}=9{\left(\frac{3}{4}\right)}^0+9{\left(\frac{3}{4}\right)}^1+9{\left(\frac{3}{4}\right)}^2+9{\left(\frac{3}{4}\right)}^3+9{\left(\frac{3}{4}\right)}^4+9{\left(\frac{3}{4}\right)}^5+9{\left(\frac{3}{4}\right)}^6+9{\left(\frac{3}{4}\right)}^7+9{\left(\frac{3}{4}\right)}^8+9{\left(\frac{3}{4}\right)}^9+9{\left(\frac{3}{4}\right)}^{10}$

  $\Rightarrow {\displaystyle \sum _{n=0}^{10}9{\left(\frac{3}{4}\right)}^n}=9\left[1+{\left(\frac{3}{4}\right)}^1+{\left(\frac{3}{4}\right)}^2+{\left(\frac{3}{4}\right)}^3+{\left(\frac{3}{4}\right)}^4+{\left(\frac{3}{4}\right)}^5+{\left(\frac{3}{4}\right)}^6+{\left(\frac{3}{4}\right)}^7+{\left(\frac{3}{4}\right)}^8+{\left(\frac{3}{4}\right)}^9+{\left(\frac{3}{4}\right)}^{10}\right]$

This is in the form of Geometric progression

With first term $a=1$

Common ratio = $r=\frac{3}{4}$

Number of terms = $n=11$

We know

Sum of terms in Geometric progression = $\frac{a\left(1-r^n\right)}{1-r}$

On substituting values

We get

  $\Rightarrow {\displaystyle \sum _{n=0}^{10}9{\left(\frac{3}{4}\right)}^n}=9\left[\frac{1\left(1-{\left(\frac{3}{4}\right)}^{11}\right)}{1-\frac{3}{4}}\right]$

  $\Rightarrow {\displaystyle \sum _{n=0}^{10}9{\left(\frac{3}{4}\right)}^n}=9\left[\frac{1-\frac{177147}{4194304}}{\frac{1}{4}}\right]$

  $\Rightarrow {\displaystyle \sum _{n=0}^{10}9{\left(\frac{3}{4}\right)}^n}=9\left[\frac{1-0.0422}{0.25}\right]$

  $\Rightarrow {\displaystyle \sum _{n=0}^{10}9{\left(\frac{3}{4}\right)}^n}=9\left[\frac{0.9577}{0.25}\right]$

  $\Rightarrow {\displaystyle \sum _{n=0}^{10}9{\left(\frac{3}{4}\right)}^n}=34.48$

Graph of given Summation is

  

From Graphing utility

Sum=9 + 6.75 + 5.063 + 3.797 +2.848 + 2.136 + 1.602 +1.201+0.901 + 0.676 +0.507 = 34.48

Therefore

By verification both are equal.