Advertisements
Advertisements
प्रश्न
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)
उत्तर
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
संबंधित प्रश्न
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.
A small terrace at a football field comprises 15 steps, each of which is 50 m long and built of solid concrete. Each step has a rise of `1/4` m and a tread of `1/2` m (See figure). Calculate the total volume of concrete required to build the terrace.
[Hint: Volume of concrete required to build the first step = `1/4 xx 1/2 xx 50 m^3`]
Find the sum of the following arithmetic progressions
`(x - y)^2,(x^2 + y^2), (x + y)^2,.... to n term`
Find the sum of first 22 terms of an A.P. in which d = 22 and a = 149.
In an A.P., if the first term is 22, the common difference is −4 and the sum to n terms is 64, find n.
Find the first term and common difference for the A.P.
5, 1, –3, –7,...
The Sum of first five multiples of 3 is ______.
A piece of equipment cost a certain factory Rs 60,000. If it depreciates in value, 15% the first, 13.5% the next year, 12% the third year, and so on. What will be its value at the end of 10 years, all percentages applying to the original cost?
Which term of the sequence 114, 109, 104, ... is the first negative term?
Write the value of a30 − a10 for the A.P. 4, 9, 14, 19, ....
Write the expression of the common difference of an A.P. whose first term is a and nth term is b.
If the sum of first n even natural numbers is equal to k times the sum of first n odd natural numbers, then k =
Q.6
Q.2
Q.6
The sum of first 14 terms of an A.P. is 1050 and its 14th term is 140. Find the 20th term.
In an A.P. sum of three consecutive terms is 27 and their products is 504. Find the terms. (Assume that three consecutive terms in an A.P. are a – d, a, a + d.)
The first term of an AP of consecutive integers is p2 + 1. The sum of 2p + 1 terms of this AP is ______.
The sum of first five multiples of 3 is ______.
The nth term of an A.P. is 6n + 4. The sum of its first 2 terms is ______.
