Question

Find the rth term of an A.P., the sum of whose first n terms is 3n² + 2n. [NCERT EXEMPLAR]

Answer 1

$$\begin{gathered} \mathrm{Let}a\mathrm{and}d\mathrm{be}\mathrm{the}\mathrm{first}\mathrm{term}\mathrm{and}\mathrm{the}\mathrm{common}\mathrm{difference}\mathrm{of}\mathrm{the}\mathrm{given}\mathrm{A}.\mathrm{P}.,\mathrm{respectively} \\ \mathrm{As},S_n=3n^2+2n \\ \mathrm{So},a=S_1=3\times 1^2+2\times 1=3+2=5\mathrm{and} \\ {S}_2=3\times 2^2+2\times 2=12+4=16 \\ \Rightarrow a+a_2=16 \\ \Rightarrow a+a+d=16 \\ \Rightarrow 2a+d=16 \\ \Rightarrow 2\times 5+d=16 \\ \Rightarrow d=16-10 \\ \Rightarrow d=6 \\ \mathrm{Now}, \\ {a}_r=a+\left(r-1\right)d \\ =5+\left(r-1\right)\times 6 \\ =5+6r-6 \\ \therefore a_r=6r-1 \end{gathered}$$