Chapter 12.3, Problem 73AYU

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 73AYU

The given sequence is arithmetic. Common difference is $-\frac{2}{3}$ and the sum of first 50 terms is 700.

Explanation of Solution

Given:

Sequence is given as $\left\{3-\frac{2}{3}n\right\}$. Considering the values of $n$ as a positive integers, the sequence becomes $\frac{7}{3},\frac{5}{3},1,\frac{1}{3},\cdots$.

The first term is $a_1=\frac{7}{3}$. The $nth$ term and the $\left(n-1\right)th$ term of the sequence are $a_n=3-\frac{2}{3}n$ and $a_{n-1}=3-\frac{2}{3}\left(n-1\right)$.

The common difference $d=3-\frac{2}{3}n-3+\frac{2}{3}n-\frac{2}{3}=-\frac{2}{3}$.

Therefore, the sequence $\left\{3-\frac{2}{3}n\right\}$ is arithmetic since the difference of successive term is 1.

Sum of $n$ terms of an arithmetic sequence $=\frac{n}{2}\left[2a_1+\left(n-1\right)d\right]$.

Sum of the first 50 terms $=\frac{50}{2}\left[2\left(\frac{7}{3}\right)+\left(50-1\right)\left(-\frac{2}{3}\right)\right]=25\times -\frac{84}{3}=700$.