Question 9
If the sum of first 7 terms of an AP is 49 and that of 17 terms is 289, find the sum of first n terms.

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Solution

$G i v e n t h a t, S_{7} = 49 S_{17} = 289 S_{7} = \frac{7}{2} [2 a + (n - 1) d] 49 = \frac{7}{2} [2 a + (7 - 1) d] 7 = (a + 3 d) a + 3 d = 7 \dots \dots (i) S i m i l a r l y, S_{17} = \frac{17}{2} [2 a + (17 - 1) d] 289 = \frac{17}{2} (2 a + 16 d) 17 = (a + 8 d) a + 8 d = 17 \dots \dots (i i) S u b t r a c t i n g e q u a t i o n (i) f r o m e q u a t i o n (i i), 5 d = 10 d = 2 F r o m e q u a t i o n (i), a + 3 (2) = 7 a + 6 = 7 a = 1 S_{n} = \frac{n}{2} [2 a + (n - 1) d] = \frac{n}{2} [2 (1) + (n - 1) \times 2] = \frac{n}{2} (2 + 2 n - 2) = \frac{n}{2} (2 n) = n^{2}$

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