Question

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

Answer 1

Let the first term of an AP be $a$ and the common difference be $d$

Given that the sum of first $7$ terms is $49$

$\Rightarrow \frac{7}{2}(2a+6d)=49$

$\Rightarrow a+3d=7$

Given that the sum of first $17$ terms is $289$

$\Rightarrow \frac{17}{2}(2a+16d)=289$

$\Rightarrow a+8d=17$

By subtracting first equation from second equation , we get $5d=10$

$\Rightarrow d=2$

So the value of $d=2$ and the value of $a=7-6=1$

Now the sum of first $n$ terms is $\frac{n}{2}(2a+(n-1\left)d\right)=\frac{n}{2}(2+(n-1)\times 2)=n^2$