Question

The sum of first 7 terms of an A.P. is 10 and that of next 7 terms is 17. Find the progression.

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have}: \\ {S}_7=10 \\ \Rightarrow \frac{7}{2}\left[2a+(7-1)d\right]=10 \\ \Rightarrow \frac{7}{2}\left[2a+6d\right]=10 \\ \Rightarrow a+3d=\frac{10}{7}...\left(\mathrm{i}\right) \\ \mathrm{Also},\mathrm{the}\mathrm{sum}\mathrm{of}\mathrm{the}\mathrm{next}\mathrm{seven}\mathrm{terms}=S_{14}-S_7=17 \\ \Rightarrow \frac{14}{2}\left[2a+\left(14-1\right)d\right]-\frac{7}{2}\left[2a+(7-1)d\right]=17 \\ \Rightarrow 7\left[2a+13d\right] \\ -\frac{7}{2}\left[2a+6d\right]=17 \\ \Rightarrow 14a+91d-7a-21d=17 \\ \Rightarrow 7a+70d=17 \\ \Rightarrow a+10d=\frac{17}{7}...\left(\mathrm{ii}\right) \\ \mathrm{From}\left(\mathrm{i}\right)\mathrm{and}\left(\mathrm{ii}\right),\mathrm{we}\mathrm{get}: \\ \frac{10}{7}-3d=\frac{17}{7}-10d \\ \Rightarrow 7d=1 \\ \Rightarrow d=\frac{1}{7} \\ \mathrm{Putting}\mathrm{the}\mathrm{value}\mathrm{in}\left(\mathrm{i}\right),\mathrm{we}\mathrm{get}: \\ a+3d\mathit{}=\frac{10}{7} \\ \Rightarrow a+\frac{3}{7}=\frac{10}{7} \\ \Rightarrow a=1 \\ \therefore a=1,d=\frac{1}{7} \end{gathered}$$
The progression thus formed is $1,\frac{8}{7},\frac{9}{7},\frac{10}{7}...$