Question

If the sum of $7$ terms of an AP is $49$ and that of $17$ terms is $289,$ find the sum of $n$ terms.

• A. $n^2$
  
• B. $2n^2$
• C. $5n^3$
• D. None of these
  

Answer 1

The correct option is A $n^2$

Let $a$ and $d$ be the first term and common difference of the given AP respectively.
Now, $S_7=49$ and $S_{17}=289$
$\Rightarrow \frac{7}{2}[2a+(7-1\left)d\right]=49$

$\therefore 7a+21d=49$

$\therefore a+3d=7$ ...(1)
Also, $\frac{17}{2}[2a+(17-1\left)d\right]=289$

$\therefore 17a+136d=289$

$\therefore a+8d=17$ ...(2)

On subtracting (1) from (2), we get
$5d=10\Rightarrow d=2$
Putting $d=2$ in (1), we get
$a=7-3\times 2=1$
Now, Required sum=$S_n=\frac{n}{2}[2a+(n-1\left)d\right]$
$S_n=\frac{n}{2}[2\times 1+(n-1)\times 2]$

$=n[1+n-1]$

$=n^2$
Hence, the required sum $=S_n=n^2$.