question_answer

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

Answer:

Given, sum of first 7 terms of an A.P \[({{S}_{7}})=49\]

and sum of first 17 terms of an A.P \[({{S}_{17}})=289\]

i.e.,                     \[{{S}_{7}}=\frac{7}{2}[2a+(7-1)d]=49\]

\[2a+6d=14\]                                                          ?(i)

And.                 \[{{S}_{17}}=\frac{17}{2}[2a+(17-1)d]=289\]

\[2a+16d=34\]                                                ?(ii)

Solving equations (i)    (ii), we get

\[_{\begin{smallmatrix}  \,\,2a\,+\,6d\,\,=\,14 \\  -\,\,\,\,\,\,\,-\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-  \\  \overline{\,\,\,\,\,\,\,\,\,\,\,10\,d\,=\,\,20} \end{smallmatrix}}^{2a\,+\,16\,d\,=\,34}\]

\[d=2\]

Putting \[d=2\] in eq. (i), we get

\[a=1\]

Hence, sum of first n term of A.P,

\[{{S}_{n}}=\frac{n}{2}[2(1)+(n-1)2]\]

\[\Rightarrow {{S}_{n}}=\frac{n}{2}[2+(n-1)2]\]

\[\Rightarrow {{S}_{n}}={{n}^{2}}\]