Question

The 9th term of an ap is equal to 7 times the second term and 12th time exceeds 5 times the third term by two find the first term and the common difference

Answer 1

$$\begin{gathered} \mathrm{Let}\mathrm{a}\mathrm{be}\mathrm{the}\mathrm{first}\mathrm{term}\mathrm{and}\mathrm{d}\mathrm{be}\mathrm{the}\mathrm{common}\mathrm{difference}. \\ \mathrm{Now},{\mathrm{a}}_9=7{\mathrm{a}}_2 \\ \Rightarrow \mathrm{a}+8\mathrm{d}=7\left(\mathrm{a}+\mathrm{d}\right) \\ \Rightarrow \mathrm{a}+8\mathrm{d}=7\mathrm{a}+7\mathrm{d} \\ \Rightarrow -6\mathrm{a}=-\mathrm{d} \\ \Rightarrow \mathrm{d}=6\mathrm{a}....\left(1\right) \\ \mathrm{Now},{\mathrm{a}}_{12}=5{\mathrm{a}}_3+2 \\ \Rightarrow \mathrm{a}+11\mathrm{d}=5\left(\mathrm{a}+2\mathrm{d}\right)+2 \\ \Rightarrow \mathrm{a}+11\mathrm{d}=5\mathrm{a}+10\mathrm{d}+2 \\ \Rightarrow \mathrm{a}+11\mathrm{d}-5\mathrm{a}-10\mathrm{d}=2 \\ \Rightarrow -4\mathrm{a}+\mathrm{d}=2 \\ \Rightarrow \mathrm{d}=2+4\mathrm{a}.....\left(2\right) \\ \mathrm{Comparing}\left(1\right)\mathrm{and}\left(2\right),\mathrm{we}\mathrm{get} \\ 6\mathrm{a}=2+4\mathrm{a} \\ \Rightarrow 2\mathrm{a}=2 \\ \Rightarrow \mathrm{a}=1 \\ \mathrm{Now},\mathrm{from}\left(1\right),\mathrm{we}\mathrm{get} \\ \mathrm{d}=6\mathrm{a}=6\times 1=6 \end{gathered}$$