Question

The sum of first 16 terms of an A.P. is 528 and sum of next 16 terms is 1552.
Find the first term and common difference of the A.P.

Answer 1

Since this is an arithmetic progression, the sequence is defined recursively by

$a_n=a_{n-1}+d$

where $d$ is the common difference and $a_n$ is the $n$th term. Solving for $a_n$ gives

$a_n=a_{n-1}+d=a_{n-2}+2d+a_{n-3}+3d=\cdots=a_1+(n-1)d$

Adding up the first 16 terms gives

$\displaystyle\sum_{n=1}^{16}(a_1+(n-1)d)=16a_1+120d$

Adding up the next 16 terms gives

$\displaystyle\sum_{n=17}^{32}(a_1+(n-1)d)=16a_1+376d$

So you have two equations with two unknowns,

$\begin{cases}16a_1+120d=528\\16a_1+376d=1552\end{cases}$

Solving gives $a_1=3$ and $d=4$.