Alg8.8

Find the sum of the first 24 terms of the sequence whose nth term is given by $a_n=3+\frac{2}{3}n\,$

Let us examine whether the given sequence is an AP.

$a_n=3+\frac{2}{3}n\,$

$a_{n+1}=3+\frac{2}{3}(n+1)=[3+\frac{2}{3}n]+\frac{2}{3}=a_n+\frac{2}{3}\,$

$a_{n+1}-a_n=\frac{2}{3}\,$ a fixed number.

Hence the given sequence is an AP with common difference $\frac{2}{3}\,$

The first term is obtained by taking $n=1\,$

Thus $a_1=3+\frac{2}{3}=\frac{11}{3}\,$

Hence,the sequence is $\frac{11}{3},\frac{13}{3},\frac{15}{3}......\,$

The sum to 24 terms is given by $S_{24}=24[\frac{11}{3}+\frac{1}{2}(24-1)\frac{2}{3}]=24[\frac{11}{3}+\frac{23}{3}]\,$

$S_{24}=24\cdot \frac{34}{3}=272\,$