If the sum. Of the first 12 terms of an A. P. is equal to the sum of the first 18 terms of the same A. P. Then find the sum of the first 30terms of the A. P
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2
Answer:
Formula,
S
n
=
2
n
[2a+(n−1)d]
Consider sum of first p terms,
S
p
=
2
p
[2a+(p−1)d]
Consider sum of first q terms,
S
q
=
2
q
[2a+(q−1)d]
Given,
S
p
=S
q
2
p
[2a+(p−1)d]=
2
q
[2a+(q−1)d]
p[2a+(p−1)d]=q[2a+(q−1)d]
2a(p−q)=d[(q−1)q−(p−1)p]
2a(p−q)=−d(p−q)[p+q−1]
2a(p−q)+d(p−q)[p+q−1]=0
(p−q)[2a+(p+q−1)d]=0
p−q
=0⇒[2a+(p+q−1)d]=0
Now,
S
p+q
=
2
p+q
[2a+(p+q−1)d]
=
2
p+q
×0
=0
Therefore the sum of p+q terms is 0.
Step-by-step explanation:
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Answer:
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