Question

how many terms of the A.P -6,-11/2,-5 _______ are needed to give the sum - 25? Explain the double answer​

Answer 1

Answer:

5 terms

Step-by-step explanation:

Given:

a1 = -6

d = a2-a1 = -11/2 - (-6) = 1/2

Sn = - 25

To find :

number of terms needed to give the sum -25.

we have,

Solution:

Sum if n terms of an A.P is given by

Sn = n/2[2a + (n-1)d]

$- 25 = \frac{n}{2} (2( - 6) + (n - 1) \frac{1}{2} ) \\ - 25 = \frac{n}{2} ( - 12 + \frac{n - 1}{2} ) \\ - 25 = \frac{n}{2} ( \frac{ - 24 + n - 1}{2} ) \\ - 25 = \frac{n}{2} ( \frac{ - 25 + n}{2} ) \\ - 25 = \frac{ - 25n + {n}^{2} }{4} \\ - 100 = - 25n + {n}^{2} \\0 = {n}^{2} - 25n + 100 \\ 0 = {n}^{2} - 20n - 5n + 100 \\ 0 = n(n - 20) - 5(n - 20) \\ 0 = (n - 5)(n - 20) \\ therefore \\ n = 5 \: \: \: or \: \: \: \: n = 20$

By taking 5 as the value of n we get the sum to be -25.

Thus, 5 terms are needed to give the sum -25.