Q. 304.8( 8 Votes )

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 :

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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