Advertisements
Advertisements
प्रश्न
If the sum of p terms of an A.P. is q and the sum of q terms is p, show that the sum of p + q terms is – (p + q). Also, find the sum of first p – q terms (p > q).
Advertisements
उत्तर
Let a be the first term and d the common difference of the given A.P.
∴ Sp = `p/2 [2a + (p - 1)d]` = q
⇒ `2a + (p - 1)d = (2q)/p` ....(i)
And Sq = `q/2[2a + (q - 1)d]` = p
⇒ `2a + (q - 1)d = (2p)/q` ....(ii)
Subtracting equation (ii) from equation (i) we get
(p – q)d = `(2q)/p - (2p)/q`
⇒ (p – q)d = `(2(q^2 - p^2))/(pq)`
⇒ (p – q)d = `(-2)/(pq) (p^2 - q^2)`
⇒ (p – q)d = `(-2)/(pq) (p + q)(p - q)`
⇒ d = `(-2)/(pq) (p + q)`
Substituting the value of d in equation (i) we get
`2a + (p - 1) [(-2(p + q))/(pq)] = (2q)/p`
⇒ 2a = `(2q)/p + (2(p - 1)(p + q))/(pq)`
⇒ a = `q/p + ((p - 1)(p + q))/(pq)`
⇒ a = `(q^2 + p^2 + pq - p - q)/(pq)`
Now Sp+q = `(p + q)/2 [2a + (p + q - 1)d]`
= `(p + q)/2 [(2q^2 + 2p^2 + 2pq - 2p - 2q)/(pq) + ((p + q - 1)[-2(p + q)])/(pq)]`
= `(p + q)/2 [(2q^2 + 2p^2 + 2pq - 2p - 2q - 2p^2 - 2pq + 2p - 2pq - 2q^2 + 2q)/(pq)]`
= `(p + q)/2 [(-2q)/(pq)]`
= `- (p + q)`
Hence proved.
APPEARS IN
संबंधित प्रश्न
How many terms of the A.P. -6 , `-11/2` , -5... are needed to give the sum –25?
If the sum of first p terms of an A.P. is equal to the sum of the first q terms, then find the sum of the first (p + q) terms.
Between 1 and 31, m numbers have been inserted in such a way that the resulting sequence is an A.P. and the ratio of 7th and (m – 1)th numbers is 5:9. Find the value of m.
If the sum of three numbers in A.P., is 24 and their product is 440, find the numbers.
if `a(1/b + 1/c), b(1/c+1/a), c(1/a+1/b)` are in A.P., prove that a, b, c are in A.P.
Shamshad Ali buys a scooter for Rs 22000. He pays Rs 4000 cash and agrees to pay the balance in annual installment of Rs 1000 plus 10% interest on the unpaid amount. How much will the scooter cost him?
A sequence is defined by an = n3 − 6n2 + 11n − 6, n ϵ N. Show that the first three terms of the sequence are zero and all other terms are positive.
Let < an > be a sequence. Write the first five term in the following:
a1 = 1, an = an − 1 + 2, n ≥ 2
Let < an > be a sequence. Write the first five term in the following:
a1 = 1 = a2, an = an − 1 + an − 2, n > 2
Find:
nth term of the A.P. 13, 8, 3, −2, ...
If the sequence < an > is an A.P., show that am +n +am − n = 2am.
How many terms are there in the A.P. 7, 10, 13, ... 43 ?
Find the sum of the following arithmetic progression :
41, 36, 31, ... to 12 terms
Find the sum of all integers between 84 and 719, which are multiples of 5.
Find the sum of all integers between 50 and 500 which are divisible by 7.
Find the sum of all those integers between 100 and 800 each of which on division by 16 leaves the remainder 7.
Find the sum of n terms of the A.P. whose kth terms is 5k + 1.
If S1 be the sum of (2n + 1) terms of an A.P. and S2 be the sum of its odd terms, then prove that: S1 : S2 = (2n + 1) : (n + 1).
The sums of n terms of two arithmetic progressions are in the ratio 5n + 4 : 9n + 6. Find the ratio of their 18th terms.
A man saved Rs 16500 in ten years. In each year after the first he saved Rs 100 more than he did in the receding year. How much did he save in the first year?
A carpenter was hired to build 192 window frames. The first day he made five frames and each day thereafter he made two more frames than he made the day before. How many days did it take him to finish the job?
Write the common difference of an A.P. whose nth term is xn + y.
If the sum of p terms of an A.P. is q and the sum of q terms is p, then the sum of p + q terms will be
In n A.M.'s are introduced between 3 and 17 such that the ratio of the last mean to the first mean is 3 : 1, then the value of n is
If the sum of n terms of an A.P., is 3 n2 + 5 n then which of its terms is 164?
In the arithmetic progression whose common difference is non-zero, the sum of first 3 n terms is equal to the sum of next n terms. Then the ratio of the sum of the first 2 n terms to the next 2 nterms is
If a1, a2, a3, .... an are in A.P. with common difference d, then the sum of the series sin d [sec a1 sec a2 + sec a2 sec a3 + .... + sec an − 1 sec an], is
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
The first and last term of an A.P. are a and l respectively. If S is the sum of all the terms of the A.P. and the common difference is given by \[\frac{l^2 - a^2}{k - (l + a)}\] , then k =
The first three of four given numbers are in G.P. and their last three are in A.P. with common difference 6. If first and fourth numbers are equal, then the first number is
The pth term of an A.P. is a and qth term is b. Prove that the sum of its (p + q) terms is `(p + q)/2[a + b + (a - b)/(p - q)]`.
In an A.P. the pth term is q and the (p + q)th term is 0. Then the qth term is ______.
The first term of an A.P.is a, and the sum of the first p terms is zero, show that the sum of its next q terms is `(-a(p + q)q)/(p - 1)`
A man accepts a position with an initial salary of Rs 5200 per month. It is understood that he will receive an automatic increase of Rs 320 in the very next month and each month thereafter. Find his salary for the tenth month
A man accepts a position with an initial salary of Rs 5200 per month. It is understood that he will receive an automatic increase of Rs 320 in the very next month and each month thereafter. What is his total earnings during the first year?
Let Sn denote the sum of the first n terms of an A.P. If S2n = 3Sn then S3n: Sn is equal to ______.
The internal angles of a convex polygon are in A.P. The smallest angle is 120° and the common difference is 5°. The number to sides of the polygon is ______.
