Question

The sum of first 8 term of an AP is 100 and sum of first 19 terms is 551. Find the AP..?

Answer 1

`2, 5, 8, 11, 14, ............................................`

Explanation:

First we will list the data we are given.

The sum of the first 8 terms of the A.P is = `100`.

The sum of the first 19 terms is = `551`.

Now Let The First term of the A.P be `a`, and the common difference be `d`.

So, According to the problem,

`color(white)(xxx)8/2[a + a + (8 - 1) d] = 100`

`rArr 4[2a + 7d] = 100`

`rArr 2a + 7d = 25`........................................(i)

And, `19/2[a + a + (19 - 1)d] = 551`

`rArr 19/2 [2a + 18d] = 551`

`rArr 19[a + 9d] = 551`

`rArr a + 9d = 29`..........................(ii)

Now, Just Solve For `a` and `d`.

FIrst, Multiply eq(ii) with `2`.

So, we get,

`2a + 18d = 58`............................(iii)

Now. Subtract eq(i) from eq(iii).

So, We get,

`color(white)(xxx)cancel(2a) + 18d cancel(- 2a) - 7d = 58 - 25`

`rArr 11d = 33`

`rArr d = 3`

Now, Substitute `d = 3` in eq(i).

So, We get,

`color(white)(xxx)2a + 7 * 3 = 25`

`rArr 2a + 21 = 25`

`rArr 2a = 25 - 21`

`rArr 2a = 4`

`rArr a = 2`

So, Now we can form the A.P.

The AP will be `a, a + d, a + 2d, a + 3d,............................,a + (n - 1)d.`

So, The Finalised A.P. is :-

`2, 5, 8, 11, 14, ............................................`