the sum of first 8 terms of an AP is 100 and the sum of its first 19 terms is 551. find the first term and the common difference of the AP.

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Solution

$W e k n o w t h a t s u m u p t o n t h t e r m S_{n t h} = \frac{n}{2} {2 a + (n - 1) d}$
here a is the first term, and d is the common difference.
$S_{8} = \frac{8}{2} {2 a + (8 - 1) d} = 100 o r 4 {2 a + 7 d} = 100 o r 2 a + 7 d = 25 (i) a n d S_{19} = \frac{19}{2} {2 a + (19 - 1) d} = 551 o r 19 (2 a + 18 d) = 1102 2 a + 18 d = 58 (i i) Hence solving (i) and (ii), we get 11 d = 33 d = 3 And putting the value of d i n (i), w e g e t a = 2 . H e n c e t h e f i r s t t e r m i s 2, and the common difference is 3 .$

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