Question

The third term of an A.P. is 7 and the seventh term exceeds three times the third term by 2. Find the first term, the common difference and the sum of first 20 terms.

Answer 1

$$\begin{gathered} \mathrm{Given}: \\ {a}_3=7,a_7-3a_3=2 \\ \mathrm{We}\mathrm{have}: \\ {a}_3=7 \\ \Rightarrow a+\left(3-1\right)d=7 \\ \Rightarrow a+2d=7...\left(i\right) \\ \mathrm{Also},a_7-3a_3=2 \\ \Rightarrow a_7-21=2\left(\mathrm{Given}\right) \\ \Rightarrow a+\left(7-1\right)d=23 \\ \Rightarrow a+6d=23...\left(\mathrm{ii}\right) \\ \mathrm{From}\left(\mathrm{i}\right)\mathrm{and}\left(\mathrm{ii}\right),\mathrm{we}\mathrm{get}: \\ 4d=16 \\ \Rightarrow d=4 \\ \mathrm{Putting}\mathrm{the}\mathrm{value}\mathrm{in}\left(\mathrm{i}\right),\mathrm{we}\mathrm{get}: \\ a+2\left(4\right)=7 \\ \Rightarrow a=-1 \\ \therefore S_{20}=\frac{20}{2}\left[2\left(-1\right)+\left(20-1\right)\left(4\right)\right] \\ \Rightarrow S_{20}=10\left[-2+76\right] \\ \Rightarrow S_{20}=10\left[74\right]=740 \\ \therefore a=-1,d=4,S_{20}=740 \end{gathered}$$