Question

Sum $3,\frac{7}{3},1\frac{2}{3}$, .... to $n$ terms.

Answer 1

We have to find the sum of terms in the given series $3,\frac{7}{3},1\frac{2}{3},........$ to $n$ terms

The given series can be written as $\frac{9}{3},\frac{7}{3},\frac{5}{3},........$ to $n$ terms

The given series can be written as $\frac{1}{3}(9,7,5,.....)$ to $n$ terms

Observe that the series $9,7,5,...$ is in Arithmetic progression with first term as $9$ and common ratio $-2$

Let the sum of given terms in the series be $L$

$\Rightarrow L=\frac{1}{3}(9+7+5+....)$ to $n$ terms

$\Rightarrow L=\frac{1}{3}\times \frac{n}{2}(2\times 9+(n-1)\times (-2\left)\right)$

$\Rightarrow L=\frac{n}{6}(20-2n)=\frac{n(10-n)}{3}$