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
An A.P. consists of 50 terms of which 3rd term is 12 and the last term is 106. Find the 29th term of the A.P.
Find the sum given below:
–5 + (–8) + (–11) + ... + (–230)
In an AP: Given a = 5, d = 3, an = 50, find n and Sn.
In an AP, given a = 2, d = 8, and Sn = 90, find n and an.
In an AP given l = 28, S = 144, and there are total 9 terms. Find a.
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.
Find the sum of the following arithmetic progressions:
a + b, a − b, a − 3b, ... to 22 terms
How many three-digit numbers are divisible by 9?
Find the sum of all multiples of 9 lying between 300 and 700.
Find the first term and common difference for the A.P.
0.6, 0.9, 1.2,1.5,...
In an A.P. the 10th term is 46 sum of the 5th and 7th term is 52. Find the A.P.
Find the A.P. whose fourth term is 9 and the sum of its sixth term and thirteenth term is 40.
The first and the last terms of an A.P. are 7 and 49 respectively. If sum of all its terms is 420, find its common difference.
The sum of first n terms of an A.P. is 5n − n2. Find the nth term of this A.P.
Two A.P.'s have the same common difference. The first term of one of these is 8 and that of the other is 3. The difference between their 30th term is
Q.2
Q.10
Which term of the AP 3, 15, 27, 39, ...... will be 120 more than its 21st term?
If the numbers n - 2, 4n - 1 and 5n + 2 are in AP, then the value of n is ______.
The nth term of an Arithmetic Progression (A.P.) is given by the relation Tn = 6(7 – n)..
Find:
- its first term and common difference
- sum of its first 25 terms
