(a) If the sum of first 7 terms of an A.P. is 49 and that of first 17 terms is 289, find the sum of its first 20 terms.

Let a be the first term and d be the common difference.

\[\begin{aligned} & \frac{7}{2}(2 \mathrm{a}+6 \mathrm{~d})=49 \\ & \mathrm{a}+3 \mathrm{~d}=7 \ldots \ldots \ldots \ldots \ldots \\ & S_{17}=289 \\ & \frac{17}{2}(2 \mathrm{a}+16 \mathrm{~d})=289 \\ & \mathrm{a}+8 \mathrm{~d}=17 \ldots \ldots \ldots \ldots . \end{aligned} \] (ii) solving (i) and (ii)

\[\begin{aligned} \mathrm{d} & =2 \& \mathrm{a}=1 \\ S_{20} & =\frac{20}{2}[2(1)+19(2)] \\ & =400 \end{aligned} \]

Let a be the first term and d be the common difference.

\[\begin{aligned} & \frac{a+9 d}{a+29 d}=\frac{1}{3} \\ & \Rightarrow \mathrm{a}=\mathrm{d} \end{aligned} \]

\[\begin{aligned} & \frac{6}{2}(2 a+5 d)=42 \\ & \Rightarrow 2 a+5 d=14 \end{aligned} \] Solving (i) and (ii)

\(\displaystyle \mathrm{a}=2\) and \(\displaystyle \mathrm{d}=2\)