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]

\begin{gathered} - 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\end{gathered}

−25=

2

n

(2(−6)+(n−1)

2

1

)

−25=

2

n

(−12+

2

n−1

)

−25=

2

n

(

2

−24+n−1

)

−25=

2

n

(

2

−25+n

)

−25=

4

−25n+n

2

−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=5orn=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.

Verified answer

Answer:

There are 12 terms taken by A.P.