Chapter 12.3, Problem 78AYU

To determine

To verify whether the given sequence is arithmetic, geometric or neither. If the sequence is arithmetic or geometric, find the common difference or common ratio accordingly and calculate the sum of the first 50 terms.

Answer to Problem 78AYU

The given sequence is geometric. Common ratio is $\frac{5}{4}$ and the sum of first 50 terms is $5\left(1-{\left(\frac{5}{4}\right)}^{50}\right)$.

Explanation of Solution

Given:

Sequence is given as $\left\{{\left(\frac{5}{4}\right)}^n\right\}$. Considering the values of $n$ as a positive integers, the sequence becomes $\frac{5}{4}, {\left(\frac{5}{4}\right)}^2, {\left(\frac{5}{4}\right)}^3, {\left(\frac{5}{4}\right)}^4,\cdots$.

The first term is $a_1=\frac{5}{4}$.

The $n$th term and the $\left(n-1\right)$th term of the sequence are $a_n={\left(\frac{5}{4}\right)}^n$ and $a_{n-1}={\left(\frac{5}{4}\right)}^{n-1}$.

The common ratio $r=\frac{{\left(\frac{5}{4}\right)}^n}{{\left(\frac{5}{4}\right)}^{n-1}}=\frac{5}{4}$.

Therefore, the sequence $\left\{{\left(\frac{5}{4}\right)}^n\right\}$ is geometric since the ratio of successive term is $\frac{5}{4}$.

Sum of $n$ terms of an geometric sequence $=S_n= a_1\frac{1-r^n}{1-r}$.

Sum of the first 50 terms $=S_n= \frac{5}{4}\times \frac{1-{\left(\frac{5}{4}\right)}^{50}}{1-\frac{5}{4}}=5\left(1-{\left(\frac{5}{4}\right)}^{50}\right)$.