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.

Answer 1

Let a be the first term and d be the common difference of first AP.
Let A be the first term and D be the common difference of second AP.
Now, a = 3; A = 8 and d = D
We know that the sum of first n terms of an AP is given by;
$S_n$ = $\frac{n}{2}$ [2a + (n - 1)d]

Let $s_{50}$ be the sum of first 50 terms of first AP.
Let $S_{50}$ be the sum of first 50 terms of second AP.
Now,
s50 − S50 = $\frac{50}{2}$ [2×3 + (50 - 1)d] - $\frac{50}{2}$ [2×8 + (50 - 1)d]

s50 − S50 = 25 [ 6 + 49d] - 25 [16+49d]

s50 − S50 = 25[6+49d] − 25[16+49d]
s50 − S50 = 25[6+49d−16−49d]

s50 − S50 =−250