Question

how many term of the series 19+17+15........... must be taken in order to get sum 91​

Answer 1

Sum of A.P ( Arithmetic Progression ) = 19+17+15....

first term = 19

Difference = a2-a1 = 17 - 19 = -2

Sum of A.P = 91

n = ?

Sn ( Sum of A.P ) = n/2 ( 2a + (n-1)d )

$91 = \frac{n}{2} (2(19) + (n - 1) - 2) \\ 91 \times 2 = n(38 - 2n + 2) \\ 182 = 38n - 2 {n}^{2} + 2n \\ 182 = 40n - 2 {n}^{2} \\ 2 {n}^{2} - 40n + 182 = 0 \\ {n}^{2} - 20n + 91 = 0 \\ {n}^{2} - 7n - 13n + 91 = 0 \\ n(n - 7) - 13(n - 7) = 0 \\ (n - 7) (n - 13) = 0 \\ (n - 7) = 0 \: or \: (n - 13) = 0 \\ n \: = 7 \: or \: 13$

If it is 13, the sum goes to negative

Hence, number of terms are 7