Chapter 12.3, Problem 75AYU

To determine

To find: Whether the sequence is arithmetic, geometric or neither, find the common difference or common ratio and also find the sum of the first fifty terms if the sequence is arithmetic or geometric.

Answer to Problem 75AYU

The sequence is geometric, common ratio is $\frac{2}{3}$ and the sum of first $50$ terms is $1.999999$ .

Explanation of Solution

Given information:

The series is ${\left(\frac{2}{3}\right)}^n$ .

Calculation:

The series can be expressed as, ${\left(\frac{2}{3}\right)}^1,{\left(\frac{2}{3}\right)}^2,{\left(\frac{2}{3}\right)}^3$

Here, $a_1=\frac{2}{3}$ , and the common ratio is $\frac{2}{3}$

Calculate the sum of first

  $n$ terms.

  $S_n=a_1\frac{1-r^n}{1-r}$

Calculate the sum of first $50$ terms.

  $\begin{array}{c}{S}_{50}=\frac{2}{3}\frac{1-{\left(\frac{2}{3}\right)}^{50}}{1-\frac{2}{3}}\\ =\frac{2}{3}\frac{1-{\left(\frac{2}{3}\right)}^{50}}{\frac{1}{3}}\\ =1.999999\end{array}$

Therefore, the sequence is geometric, common ratio is $\frac{2}{3}$ and the sum of first

  $50$ terms is $1.999999$ .