if the sum of first 7 terms of an AP is 49 and that 17 terms is 289. find the sum of first 'n' terms
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Answer:
Given, Sum of 7 terms of an A.P. is 49 ⟹ S7 = 49 And, sum of 17 terms of an A.P. is 289 ⟹ S17 = 289 Let the first term of the A.P be a and common difference as d. And, we know that the sum of n terms of an A.P is Sn =n2n2[2a + (n − 1)d] So, S7 = 49 = 7272[2a + (7 – 1)d] = 7272 [2a + 6d] = 7[a + 3d] ⟹ 7a + 21d = 49 a + 3d = 7 ….. (i) Similarly, S17 = 172172[2a + (17 – 1)d] = 172172 [2a + 16d] = 17[a + 8d] ⟹ 17[a + 8d] = 289 a + 8d = 17 ….. (ii) Now, subtracting (i) from (ii), we have a + 8d – (a + 3d) = 17 – 7 5d = 10 d = 2 Putting d in (i), we find a a + 3(2) = 7 a = 7 – 6 = 1 So, for the A.P: a = 1 and d = 2 For the sum of n terms is given by, Sn = n2n2[2(1) + (n − 1)(2)] = n2n2[2 + 2n – 2] = n2n2[2n] = n2 Therefore, the sum of n terms of the A.P is given by n2.Read more on Sarthaks.com - https://www.sarthaks.com/655565/if-the-sum-of-7-terms-of-an-a-p-is-49-and-that-of-17-terms-is-289-find-the-sum-of-n-terms