# 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.

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

Let d1 and d2 be the common difference of the respective APs.

Given: d1 = d2 and a1 = 3, a2 = 8

To find: Difference between the sums of their first 50 terms.

i.e. to find: (S2)50 - (S1)50

where (S1)50 denotes the sum of first 50 terms of first AP and (S2)50

denotes the sum of first 50 terms of second AP.

Now, consider (S1)50 = (50/2)[2a1 + (50 - 1)d1]

= 25 × [2(3) + 49 × d1]

= 25[6 + 49d1]

= 150 + 1225d1

Now, consider (S2)50 = (50/2)[2a2 + (50 - 1)d2]

= 25 × [2(8) + 49 × d2]

= 25[16 + 49d1]

= 400 + 1225d2

Now, (S2)50 - (S1)50 = 400 + 1225d2 - (150 + 1225d2)

= 400 - 150 ( d1 = d2 )

= 250

(S2)50 - (S1)50 = 250

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