# Two APs have the same common difference.

Question:

Two APs have the same common difference. If the first terms of these APs be 3 and 8 respectively, find the difference between the sums of their first 50 terms.

Solution:

Let a1 and a2 be the first terms of the two APs.

Here, a= 8 and a2 = 3

Suppose be the common difference of the two APs.

Let $S_{50}$ and $S_{50}^{\prime}$ denote the sums of their first 50 terms.

$\therefore S_{50}-S_{50}^{\prime}=\frac{50}{2}\left[2 a_{1}+(50-1) d\right]-\frac{50}{2}\left[2 a_{2}+(50-1) d\right]$

$=25(2 \times 8+49 d)-25(2 \times 3+49 d)$

$=25 \times(16-6)$

$=250$

Hence, the required difference between the two sums is 250.