Question

find the 1st term of the geometric progression in which s8=510, r=2

Answer 1

$$\begin{gathered} \mathrm{Let}a\mathrm{be}\mathrm{the}\mathrm{first}\mathrm{term}\mathrm{and}r\mathrm{be}\mathrm{the}\mathrm{common}\mathrm{ratio}\mathrm{of}\mathrm{the}\mathrm{given}\mathrm{geometric}\mathrm{progression}\mathrm{then}, \\ \mathrm{Sum}\mathrm{of}n\mathrm{terms}\mathrm{of}\mathrm{a}\mathrm{GP}\mathrm{is}\mathrm{written}\mathrm{as}{\mathrm{S}}_n=\frac{a\left(r^n-1\right)}{\left(r-1\right)} \\ \mathrm{Putting}r=2\mathrm{and}n=8,\mathrm{we}\mathrm{get}, \\ {S}_8=\frac{a\left(2^8-1\right)}{\left(2-1\right)} \\ \Rightarrow 510=\frac{a\left(256-1\right)}{1}\left[\mathbf{as}\mathbf{}{\mathbf{S}}_{\mathbf{8}}\mathbf{}\mathbf{=}\mathbf{}\mathbf{510}\right] \\ \Rightarrow 510=a\times 255 \\ \Rightarrow a=\frac{510}{255}=2 \\ \mathbf{Therefore}\mathbf{,}\mathbf{}\mathbf{first}\mathbf{}\mathbf{term}\mathbf{}\mathbf{of}\mathbf{}\mathbf{given}\mathbf{}\mathbf{geometric}\mathbf{}\mathbf{progression}\mathbf{}\mathbf{is}\mathbf{}\mathbf{2} \end{gathered}$$