Question

If the sum of 1st 7 terms of an AP is 49 and that of 17 terms is 289 find the sum of 1st n term​

Answer 1

ɢɪᴠᴇɴ:-

★Sum of 1st 7 terms is = 49

★Sum of 17 terms is = 289

ᴛᴏ ғɪɴᴅ:-

★Sum of 1st n terms

sᴏʟᴜᴛɪᴏɴ:-

We know,

$\boxed{\sf{S_{n}=\frac{n}{2}[2a+(n-1)d]}}$

☞Hence,

according to the 1st condition ____⋆

$\sf{\therefore{S_{7}=49}}$

$\sf{\therefore{S_{7}=\frac{7}{2}[2a+(7-1) \times d]}}$

$⟹7=a+3d$

...............eq(1)

☞According to the 2nd condition ____⋆

$\sf{\therefore{S_{17}=289}}$

$\sf{\therefore{S_{17}=\frac{17}{2}[2a+(17-1) \times d]}}$

$⟹17 = a \: + 8d$

.............eq(2)

Now,

eq(2) -eq(1)

⟹ 17 -7 = a +8d -a +3d

⟹ d =2

Putting the value of d in eq( 1), we get

a =1

Therefore,

$\sf{\therefore{S_{n}=\frac{n}{2}[2 \times 1+(n-1) \times 2]}} \\ \\ =\frac{n}{2} \times 2n \\ \\ = {n}^{2}$

$\sf\blue{\tt{\therefore{The \ sum \ of \ first \ n \ terms \ is \ {n}^{2} .}}}$

$\bold\blue{Thanks...!! }$

Answer 2

Please refer the attachments for the answer.

$\bigstar\mathcal\pink{Hope-it-helps-u}$