Question

In an Arithmetic Progression the sum of first 10 terms is 175 and the sum of the

next 10 terms is 475. Find the Arithmetic Progression

Answer 1

Let a be the first term and d be the common difference of the required A.P
We know that, sum of first n terms in an A.P is given by
$$\begin{gathered} S_n=\frac{n}{2}\left[2a+\left(n-1\right)d\right] \\ \mathrm{So},S_{10}=\frac{10}{2}\left[2a+9d\right] \\ \Rightarrow 175=5\left[2a+9d\right] \\ \Rightarrow 2a+9d=35..............\left(1\right) \\ \mathrm{Now},S_{20}=\frac{20}{2}\left[2a+19d\right] \\ \mathrm{So},\mathrm{sum}\mathrm{of}\mathrm{last}10\mathrm{terms}=S_{20}-S_{10} \\ \Rightarrow 475=10\left(2a+19d\right)-175\left[\mathrm{as},S_{10}=175\right] \\ \Rightarrow 10\left(2a+19d\right)=475+175 \\ \Rightarrow 10\left(2a+19d\right)=650 \\ \Rightarrow 2a+19d=65.............\left(2\right) \\ \mathrm{Subtracting}\left(1\right)\mathrm{from}\left(2\right),\mathrm{we}\mathrm{get}, \\ \left(2a+19d\right)-\left(2a+9d\right)=65-35 \\ \Rightarrow 10d=30 \\ \Rightarrow \mathit{d}\mathbf{}\mathbf{=}\mathbf{}\mathbf{3} \end{gathered}$$

$$\begin{gathered} \mathrm{Put}d=3\mathrm{in}\left(1\right),\mathrm{we}\mathrm{get}, \\ 2a+9\times 3=35 \\ \Rightarrow 2a+27=35 \\ \Rightarrow 2a=35-27 \\ \Rightarrow 2a=8 \\ \Rightarrow \mathbf{}\mathit{a}\mathbf{}\mathbf{=}\mathbf{}\mathbf{4} \end{gathered}$$

$$\begin{gathered} \mathrm{So}\mathrm{the}\mathrm{required}\mathrm{A}.\mathrm{P}\mathrm{is}:a,\left(a+d\right),\left(a+2d\right),...... \\ \mathrm{that}\mathrm{is}\mathrm{required}\mathrm{A}.\mathrm{P}\mathrm{is}4,7,10,...... \end{gathered}$$