Advertisements
Advertisements
प्रश्न
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)`
Advertisements
उत्तर
Given that a1 = a and Sp = 0
Sum of next q terms of the given A.P. = Sp+q – Sp
∴ `"S"_(p + q) = (p + q)/2[2a + (p + q - 1)d]`
And Sp = `p/2 [2a + (p - 1)d]` = 0
⇒ 2a + (p – 1)d = 0
⇒ (p – 1)d = – 2a
⇒ d = `(-2a)/(p - 1)`
Sum of next q terms = Sp+q – Sp
= `(p + q)/2[2a + (p + q- 1)d]` = 0
= `(p + q)/2[2a + (p + q - 1) ((-2a)/(p - 1))]`
= `(p + q)/2[2a + ((p - 1)(-2a))/(p - 1) - (2aq)/(p - 1)]`
= `(p + q)/2[2a - 2a - (2aq)/(p - 1)]`
= `((p + q))/2((-2aq)/(p - 1))`
= `(-a(p + q)q)/(p - 1)`
Hence, the required sum = `(-a(p + q)q)/(p - 1)`
APPEARS IN
संबंधित प्रश्न
Find the sum of all natural numbers lying between 100 and 1000, which are multiples of 5.
In an A.P, the first term is 2 and the sum of the first five terms is one-fourth of the next five terms. Show that 20th term is –112.
The ratio of the sums of m and n terms of an A.P. is m2: n2. Show that the ratio of mth and nthterm is (2m – 1): (2n – 1)
Insert five numbers between 8 and 26 such that the resulting sequence is an A.P.
If the sum of three numbers in A.P., is 24 and their product is 440, find the numbers.
Let the sum of n, 2n, 3n terms of an A.P. be S1, S2 and S3, respectively, show that S3 = 3 (S2– S1)
Find the sum of integers from 1 to 100 that are divisible by 2 or 5.
A manufacturer reckons that the value of a machine, which costs him Rs 15625, will depreciate each year by 20%. Find the estimated value at the end of 5 years.
Let < an > be a sequence defined by a1 = 3 and, an = 3an − 1 + 2, for all n > 1
Find the first four terms of the sequence.
Find:
nth term of the A.P. 13, 8, 3, −2, ...
Which term of the A.P. 84, 80, 76, ... is 0?
How many terms are there in the A.P.\[- 1, - \frac{5}{6}, -\frac{2}{3}, - \frac{1}{2}, . . . , \frac{10}{3}?\]
The sum of 4th and 8th terms of an A.P. is 24 and the sum of the 6th and 10th terms is 34. Find the first term and the common difference of the A.P.
How many numbers are there between 1 and 1000 which when divided by 7 leave remainder 4?
If < an > is an A.P. such that \[\frac{a_4}{a_7} = \frac{2}{3}, \text { find }\frac{a_6}{a_8}\].
Find the sum of first n natural numbers.
Find the sum of all natural numbers between 1 and 100, which are divisible by 2 or 5.
Show that the sum of all odd integers between 1 and 1000 which are divisible by 3 is 83667.
If the 5th and 12th terms of an A.P. are 30 and 65 respectively, what is the sum of first 20 terms?
Find the sum of all two digit numbers which when divided by 4, yields 1 as remainder.
How many terms of the A.P. −6, \[- \frac{11}{2}\], −5, ... are needed to give the sum −25?
If \[\frac{1}{a}, \frac{1}{b}, \frac{1}{c}\] are in A.P., prove that:
\[\frac{b + c}{a}, \frac{c + a}{b}, \frac{a + b}{c}\] are in A.P.
If a2, b2, c2 are in A.P., prove that \[\frac{a}{b + c}, \frac{b}{c + a}, \frac{c}{a + b}\] are in A.P.
If \[\frac{b + c}{a}, \frac{c + a}{b}, \frac{a + b}{c}\] are in A.P., prove that:
\[\frac{1}{a}, \frac{1}{b}, \frac{1}{c}\] are in A.P.
If \[\frac{b + c}{a}, \frac{c + a}{b}, \frac{a + b}{c}\] are in A.P., prove that:
bc, ca, ab are in A.P.
Insert five numbers between 8 and 26 such that the resulting sequence is an A.P.
If n arithmetic means are inserted between 1 and 31 such that the ratio of the first mean and nth mean is 3 : 29, then the value of n is
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 =
If in an A.P., Sn = n2p and Sm = m2p, where Sr denotes the sum of r terms of the A.P., then Sp is equal to
Mark the correct alternative in the following question:
\[\text { If in an A . P } . S_n = n^2 q \text { and } S_m = m^2 q, \text { where } S_r \text{ denotes the sum of r terms of the A . P . , then }S_q \text { equals }\]
Mark the correct alternative in the following question:
Let Sn denote the sum of first n terms of an A.P. If S2n = 3Sn, then S3n : Sn is equal to
If a, b, c are in G.P. and a1/x = b1/y = c1/z, then xyz are in
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)]`.
Find the sum of first 24 terms of the A.P. a1, a2, a3, ... if it is known that a1 + a5 + a10 + a15 + a20 + a24 = 225.
If a, b, c, d are four distinct positive quantities in A.P., then show that bc > ad
If the sum of n terms of an A.P. is given by Sn = 3n + 2n2, then the common difference of the A.P. is ______.
Let 3, 6, 9, 12 ....... upto 78 terms and 5, 9, 13, 17 ...... upto 59 be two series. Then, the sum of the terms common to both the series is equal to ______.
If a1, a2, a3, .......... are an A.P. such that a1 + a5 + a10 + a15 + a20 + a24 = 225, then a1 + a2 + a3 + ...... + a23 + a24 is equal to ______.
