How many terms of the A.P. , -6, - , −5, . are needed to give the sum – 25? Explain double answer.
Answers
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.
Answer:
There are 12 terms taken by A.P.
