Question

Sum of first $n$ positive terms of an A.P. is given by $S_{n}=\left(1+2 T_{n}\right)\left(1-T_{n}\right)$. If the value of $T_{2}^{2}$ is $\frac{\sqrt{2}-1}{ k \sqrt{2}}$ then find $k$.

Answer 1

$T _{1}= S _{1}=\left(1+2 T _{1}\right)\left(1- T _{1}\right)$
$\Rightarrow T _{1}=1 / \sqrt{2}\left( T _{1}>0\right)$
Now, $T _{1}+ T _{2}= S _{2}$
$=\left(1+2 T _{2}\right)\left(1- T _{2}\right)$
$\Rightarrow 2 T _{2}^{2}=1- T _{1}=1-\frac{1}{\sqrt{2}}=\frac{\sqrt{2}-1}{\sqrt{2}}$
or $T _{2}^{2}=\frac{\sqrt{2}-1}{2 \sqrt{2}}$