Question

If the sum of positive terms of the series $10+9 \frac{4}{7}+9 \frac{1}{7}+.$ is $\frac{k}{7}$ then find value of $k$.

Answer 1

$10+\frac{67}{7}+\frac{64}{7}+\ldots \ldots$
$=\frac{70}{7}+\frac{67}{7}+\frac{64}{7} \ldots \ldots+ T _{ n }$
let $T _{ n }<0$
$\Rightarrow \frac{70}{7}+( n -1) \cdot\left(\frac{-3}{7}\right)<0$
$10-\frac{3}{7} n +\frac{3}{7}<0$
$\frac{3 n}{7}>\frac{73}{7}$
$3 n >73$
$n >24.3 \ldots \ldots$
$\because n \in N $
$\Rightarrow n =25,26, \ldots \ldots$
$\Rightarrow T _{25}=$ first - ve term
$\therefore$ Sum of positive terms $= S _{24}$
$=\frac{24}{2}\left[2 \times 10+(24-1)\left(\frac{-3}{7}\right)\right]$
$=12 \times\left[20-\frac{69}{7}\right]$
$=\frac{12 \times 71}{7}=\frac{852}{7}$