If S_n=2 3 (2^n-1 ), find T_4. - Vidyadip

Question

If $S_n=\frac{2}{ 3 }\left(2^n-1\right)$, find $T_4$.

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Answer

$S_n=\frac{2}{3}\left(2^n-1\right)$
Now, $T_{n+1}=S_{n+1}-S_n$
$=\frac{2}{3}\left[2^{n+1}-1\right]-\frac{2}{3}\left[2^n-1\right]$
$=\frac{2}{3}\left[2^{n+1}-1-2^n+1\right]$
$=\frac{2}{3}\left[2^n(2-1)\right]$
$\therefore T_{n+1}=\frac{2}{3} \times 2^n$
We have to find 4 th term. Therefore, put $n =3$
$T _4=\frac{2}{3} \times 2^3$
$=\frac{2}{3} \times 8=\frac{16}{3}$

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