Chapter 12, Problem 49RE

To determine

Calculate whether the expression is a partial sum of an arithmetic or geometric sequence and also find the sum for the given sequence $1+0.9+{\left(0.9\right)}^2+\mathrm{......}{\left(0.9\right)}^5$ .

Answer to Problem 49RE

The expression is a partial sum of geometric sequence and also the sum for the given sequence $1+0.9+{\left(0.9\right)}^2+\mathrm{......}{\left(0.9\right)}^5$ is $S_n=4.68559$

Explanation of Solution

Given information:

It is given in the question that, the sequence is $1+0.9+{\left(0.9\right)}^2+\mathrm{......}{\left(0.9\right)}^5$ .

Formula used:

  $\begin{array}{l}{S}_n=a\frac{1-r^n}{1-r}\&a_n=a{r}^{n-1}\\ where,\\ aisthefirstterm\\ nisthenumberofterm\\ risthecommanratio\\ a_{n}isthenthterm\end{array}$

Calculation:

The given sequence $1+0.9+{\left(0.9\right)}^2+\mathrm{......}{\left(0.9\right)}^5$ means,

Here, in this sequence the ration of two consecutive terms are same that is $r=0.9$

So the given sequence is in geometric sequence.

Here, $a=1\&r=0.9$

Let $a_n={\left(0.9\right)}^5$ by using formula $a_n=a{r}^{n-1}$ .

  $\begin{array}{l}{a}_n={\left(0.9\right)}^5\\ a{r}^{n-1}={\left(0.9\right)}^5\\ 1{\left(0.9\right)}^{n-1}={\left(0.9\right)}^5\\ onequatingbothside\\ n-1=5\\ n=6\end{array}$

For finding the sum for the given sequence $1+0.9+{\left(0.9\right)}^2+\mathrm{......}{\left(0.9\right)}^5$ use the formula $S_n=a\frac{1-r^n}{1-r}$

Now, $a=1,d=0.9\&n=6$

  $\begin{array}{l}{S}_n=a\frac{1-r^n}{1-r}\\ =1*\frac{1-{\left(0.9\right)}^6}{1-0.9}\\ =1*\frac{1-0.5314}{0.1}\\ =\frac{0.468559}{0.1}\\ =4.68559\end{array}$

Finally, $S_n=4.68559$ .