Question

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

the AP.

Answer 1

$S_n=\frac{n}{2}[2a+(n-1\left)d\right]$
The sum of the first 8 terms of an AP is 100
$$\begin{gathered} \frac{8}{2}[2a+7d]=100; \\ 4(2a+7d)=100; \\ 2a+7d=25...\left(1\right) \end{gathered}$$
the sum of its first 19 terms is 551
$$\begin{gathered} \frac{19}{2}[2a+18d]=551; \\ \frac{19}{2}2(a+9d)=551; \\ 19(a+9d)=551; \\ a+9d=29; \\ 2a+18d=58...\left(2\right) \end{gathered}$$
(2)-(1) gives,
11d=33
d=3

From (1),
2a+7(3)=25
2a+21=25
2a=4
a=2
So a=2 and d=3
So AP is
2, 5, 8,11,14,....