Question

Sum of first $n$ positive terms of an A.P. is given by $S_n=\left(1+2T_n\right)\left(1-T_n\right)$. If the value of $T_2^2$ is $\frac{\sqrt{2}-1}{k\sqrt{2}}$ then find $k$.

Answer 1

Answer: 2.00

$T_1=S_1=\left(1+2T_1\right)\left(1-T_1\right)$
$\Rightarrow T_1=1/\sqrt{2}\left(T_1>0\right)$
Now, $T_1+T_2=S_2$
$=\left(1+2T_2\right)\left(1-T_2\right)$
$\Rightarrow 2T_2^2=1-T_1=1-\frac{1}{\sqrt{2}}=\frac{\sqrt{2}-1}{\sqrt{2}}$
or $T_2^2=\frac{\sqrt{2}-1}{2\sqrt{2}}$