The $99^{ ext {th }}$ term of the series $2+7+14+23 +34+\ldots$ is

Question

The $99^{	ext {th }}$ term of the series $2+7+14+23 +34+\ldots$ is

Professor PC Thomas Classes · Answer

$ \begin{aligned} & \mathrm{S}_{\mathrm{n}}=2+7+14+23+\ldots+\mathrm{t}_{\mathrm{n}-1}+\mathrm{t}_{\mathrm{n}} \ & \quad \mathrm{~S}_{\mathrm{n}}=\quad 2+7+14+\ldots \mathrm{t}_{\mathrm{n}-2}+\mathrm{t}_{\mathrm{n}-1}+\mathrm{t}_{\mathrm{n}} \ & \quad 0=2+[5+7+9+\ldots(\mathrm{n}-1) 	ext { term }]-\mathrm{t}_{\mathrm{n}} \ & \mathrm{t}_{\mathrm{n}}-2=\frac{\mathrm{n}-1}{2}[2 	imes 5+(\mathrm{n}-2) 2]=(\mathrm{n}-1)(\mathrm{n}+3) \ & \mathrm{n}=99 \ & \mathrm{t}_{99}=2+98 	imes 102=9998 \end{aligned} $