Advertisements
Advertisements
Question
If the sum of first p terms of an A.P. is equal to the sum of first q terms then show that the sum of its first (p + q) terms is zero. (p ≠ q)
Advertisements
Solution
Given that `S_p=S_n`
We know,
`S_n = n/2[2a + (n - 1)d]`
Let a be the first term of the AP and d be the common difference
`p/2(2a+(p-1)d)=q/2(2a+(q-1)d)`
`p(2a+(p-1)d)=q(2a+(q-1)d)`
`2ap+(p-1)dp=2aq+(q-1)dq`
`2ap-2aq+(p-1)dp-(q-1)dq=0`
`2a(p-q)+d(p^2-p-q^2+q)=0`
`2a(p-q)+d[p^2-q^2-1(p-q)]=0`
`2a(p-q)+d[(p+q)(p-q)-1(p-q)]=0`
`2a(p-q)+d(p-q)[p+q-1]=0`
p ≠ q
p - q ≠ 0
Dividing throughout by (p - q),since p ≠ q.
`2a+(p+q-1)d=0` ... (i)
Using eq (i)
`S_(p+q)=(p+q)/2(2a+(p+q-1)d)`
`S_(p+q)=(p+q)/2xx0`
`S_(p+q)=0`
Hence, the sum of its first (p + q) terms is zero.
APPEARS IN
RELATED QUESTIONS
A sum of Rs 700 is to be used to give seven cash prizes to students of a school for their overall academic performance. If each prize is Rs 20 less than its preceding prize, find the value of each of the prizes.
Three numbers are in A.P. If the sum of these numbers is 27 and the product 648, find the numbers.
Find the sum of all integers between 50 and 500, which are divisible by 7.
Determine the A.P. Whose 3rd term is 16 and the 7th term exceeds the 5th term by 12.
Find the sum of first n even natural numbers.
The first term of an AP is p and its common difference is q. Find its 10th term.
If (2p – 1), 7, 3p are in AP, find the value of p.
Find the sum of first n terms of an AP whose nth term is (5 - 6n). Hence, find the sum of its first 20 terms.
Find the 25th term of the AP \[- 5, \frac{- 5}{2}, 0, \frac{5}{2}, . . .\]
In an A.P. 17th term is 7 more than its 10th term. Find the common difference.
Choose the correct alternative answer for the following question .
If for any A.P. d = 5 then t18 – t13 = ....
In an A.P., the first term is 22, nth term is −11 and the sum to first n terms is 66. Find n and d, the common difference
If four numbers in A.P. are such that their sum is 50 and the greatest number is 4 times, the least, then the numbers are
If Sn denote the sum of the first n terms of an A.P. If S2n = 3Sn, then S3n : Sn is equal to
Q.16
Q.20
Find whether 55 is a term of the A.P. 7, 10, 13,... or not. If yes, find which term is it.
In an A.P. a = 2 and d = 3, then find S12
If the first term of an AP is –5 and the common difference is 2, then the sum of the first 6 terms is ______.
k + 2, 2k + 7 and 4k + 12 are the first three terms of an A.P. The first term of this A.P. is ______.
