Mathematics - Sequence & Series Question with Solution | TestHub

$S=2+5+12+31+86+\dots +t_n$

$S=2+5+12+31+\dots +t_{n-1}+t_n$

$0=2+3+7+19+55+\dots nterms-t_n$

$\Rightarrow t_n=2+3+7+19+55+\dots +t_n$

$\Rightarrow t_n=2+3+7+19+\dots +t_{n-1}^{\prime }+t_n^{\prime }$

Subtracting:

$$\begin{array}{ll}& 0=2+1+4+12+36+\dots nterms-t_n^{\prime }\\ \Rightarrow & t_n^{\prime }=3+[4+12+36+\dots (n-2)terms]\\ \Rightarrow & t_n^{\prime }=3+\frac{4(3^{n-2}-1)}{3-1}\\ \Rightarrow & t_n^{\prime }=2\cdot 3^{n-2}+1,(n\ge 2)\\ Now& t_n=\Sigma t_n^{\prime }=2{\sum }_{n=2}^n{3}^{n-2}+{\sum }_{n=2}^{n}1+2\\ Now& \begin{array}{l}{t}_n=3^{n-1}+n\\ S_n=\Sigma t_n=\Sigma 3^{n-1}+\Sigma n\end{array}\\ =& \frac{3^n-1+n^2+n}{2}\\ t_n=3^{n-1}+n,S_n=\frac{3^n-1+n^2+n}{2}& \text{placeholder}\end{array}$$