Question

If the sum of n terms of an A.P., is 3 n² + 5 n then which of its terms is 164?
(a) 26th
(b) 27th
(c) 28th
(d) none of these.

Answer 1

(b) 27th

$$\begin{gathered} S_n=3n^2+5n \\ {S}_1=3{\left(1\right)}^2+5\left(1\right)=8 \\ \therefore a_1=8 \\ {S}_2=3{\left(2\right)}^2+5\left(2\right)=22 \\ \therefore a_1+a_2=22 \\ \Rightarrow a_2=14 \\ \mathrm{Common}\mathrm{difference},d=14-8=6 \\ \mathrm{Also},a_n=164 \\ \Rightarrow a+\left(n-1\right)d=164 \\ \Rightarrow 8+\left(n-1\right)6=164 \\ \Rightarrow n=27 \end{gathered}$$