Question

the sum of first 8 terms of an AP is 100 and sum of first 19 terms is 551 find AP.​

Answer 1

Sum of 1st 8 terms of Ap is 100

a8 = 100

a + 7d = 100 ===> eqn 1

The sum of 1st 19th terms is 551

a19 = 551

a + 18d = 551 ===> eqn 2

From eqn 1 and 2

a + 7d = 100

a + 18d = 551

- - -

-11d = -451

d = -451 / -11

d = 41

"d" value substitute in eqn 1

a + 7d = 100

a + 7 × 41 = 100

a + 287 = 100

a = 100 - 287

a = -187

The first three terms of Ap are

a , a+d , a + 2d

a = -187

a + d = -187 + 41 = 146

a + 2d = -187 + 2 × 41 = -187 + 82 = 105

Answer 2

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,

$\frac{8}{2} \: (a + a+( 8 - 1)d) = 100 \\ \\ = > 4(2a + 7d) = 100 \\ \\ = > 2a + 7d = 25..........(1) \\ \\ and. \frac{19}{2} (a + a + (19 - 1)d = 551 \\ \\ = > \frac{19}{2} (2a + 18d) = 551 \\ \\ = > 19(a + 9d) = 551 \\ \\ = >( a + 9d) = 29 .......(2)$

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,

2a + 18d-2a-7d=58 -25

⇒11d = 33

⇒d=3

Now, Substitute

d=3

in eq(i).

So, We get,

2a +7 . 3 =25

⇒2a + 21 = 25

⇒2a= 25 −21

⇒2a=4

⇒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...........✔️✔️✔️

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