Question

Find the second term and nth term of an A.P. whose 6th term is 12 and the 8th term is 22.

Answer 1

Given:
$$\begin{gathered} a_6=12 \\ \Rightarrow a+\left(6-1\right)d=12 \\ \Rightarrow a+5d=12...\left(\mathrm{i}\right) \\ {a}_8=22 \\ \Rightarrow a+\left(8-1\right)d=22 \\ \Rightarrow a+7d=22...\left(\mathrm{ii}\right) \\ \mathrm{Solving}\left(\mathrm{i}\right)\mathrm{and}\left(\mathrm{ii}\right),\mathrm{we}\mathrm{get}: \\ 2d=10 \\ \Rightarrow d=5 \\ \mathrm{Putting}\mathrm{the}\mathrm{value}\mathrm{of}d\mathrm{in}\left(\mathrm{i}\right),\mathrm{we}\mathrm{get}: \\ a+5\times 5=12 \\ \Rightarrow a=12-25=-13 \\ \therefore a_2=a+\left(2-1\right)d=a+d=-13+5=-8 \\ \mathrm{Also},a_n=a+(n-1)d \\ =-13+(n-1)5 \\ =-13+5n-5 \\ =5n-18 \end{gathered}$$