How many terms of an A P: -6, -11/2, -5 are needed to - Maths - Arithmetic Progressions

Question

How many terms of an A P: -6, -11/2, -5 are needed to get the
sum -25 ?

Shridhar Kaushik · Accepted Answer

The given A P is $- 6, \frac{- 11}{2}, - 5,$ Here first term, a = â€“6, $d = \frac{- 11}{2} - (- 6) = \frac{- 11}{2} + 6 = \frac{1}{2}$ Let sum of n terms of the given A P is â€“25 $∴ S_{n} = - 25 \Rightarrow \frac{n}{2} {2 a + (n - 1) d} = - 25 \Rightarrow n {2 a + (n - 1) d} = - 25 \times 2 \Rightarrow n {2 \times (- 6) + (n - 1) (\frac{1}{2})} = - 50 \Rightarrow - 12 n + \frac{(n^{2} - n)}{2} = - 50 \Rightarrow - 24 n + n^{2} - n = - 100 \Rightarrow n^{2} - 25 n + 100 = 0 \Rightarrow n^{2} - 20 n - 5 n + 100 = 0 \Rightarrow n (n - 20) - 5 (n - 20) = 0 \Rightarrow (n - 20) (n - 5) = 0 \Rightarrow n = 20 or n = 5$ Hence sum of first 5 terms and sum of first 20 terns will be â€“25

How many terms of an A.P: -6, -11/2, -5 ...... are needed to get the sum -25 ?