Advertisements
Advertisements
Question
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)]`.
Advertisements
Solution
Let A be the first term and D be the common difference of the A.P.
It is given that tp = a
⇒ A + (p – 1) D = a .....(1)
tq = b
⇒ A + (q – 1) D = b .....(2)
Subtracting (2) from (1), we get
(p – 1 – q + 1) D = a – b
⇒ D = `(a - b)/(p - q)` .....(3)
Adding (1) and (2), we get
2A + (p + q – 2) D = a + b
⇒ 2A + (p + q – 1) D = a + b + D
⇒ 2A + (p + q – 1) D = `a + b"+ (a - b)/(p - q)` ....(4)
Now Sp+q = `(p + q)/2 [2"A" + (p + q - 1) "D"]`
= `(p + q)/2[a + b + (a - b)/(p - q)]` ...[(Using ...(3) and (4)]
APPEARS IN
RELATED QUESTIONS
Find the sum of all natural numbers lying between 100 and 1000, which are multiples of 5.
The sums of n terms of two arithmetic progressions are in the ratio 5n + 4: 9n + 6. Find the ratio of their 18th 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.
Find the sum of integers from 1 to 100 that are divisible by 2 or 5.
The pth, qth and rth terms of an A.P. are a, b, c respectively. Show that (q – r )a + (r – p )b + (p – q )c = 0
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
Show that the following sequence is an A.P. Also find the common difference and write 3 more terms in case.
9, 7, 5, 3, ...
Find:
nth term of the A.P. 13, 8, 3, −2, ...
The first term of an A.P. is 5, the common difference is 3 and the last term is 80; find the number of terms.
If the nth term of the A.P. 9, 7, 5, ... is same as the nth term of the A.P. 15, 12, 9, ... find n.
Find the second term and nth term of an A.P. whose 6th term is 12 and the 8th term is 22.
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.
If < an > is an A.P. such that \[\frac{a_4}{a_7} = \frac{2}{3}, \text { find }\frac{a_6}{a_8}\].
The sum of three numbers in A.P. is 12 and the sum of their cubes is 288. Find the numbers.
If the sum of three numbers in A.P. is 24 and their product is 440, find the numbers.
Find the sum of the following arithmetic progression :
a + b, a − b, a − 3b, ... to 22 terms
Find the sum of the following serie:
101 + 99 + 97 + ... + 47
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 a, b, c is in A.P., then show that:
b + c − a, c + a − b, a + b − c 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.
Insert five numbers between 8 and 26 such that the resulting sequence is an A.P.
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?
The income of a person is Rs 300,000 in the first year and he receives an increase of Rs 10000 to his income per year for the next 19 years. Find the total amount, he received in 20 years.
In a potato race 20 potatoes are placed in a line at intervals of 4 meters with the first potato 24 metres from the starting point. A contestant is required to bring the potatoes back to the starting place one at a time. How far would he run in bringing back all the potatoes?
Write the common difference of an A.P. whose nth term is xn + y.
If the sum of n terms of an AP is 2n2 + 3n, then write its nth term.
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
If the sum of n terms of an A.P. be 3 n2 − n and its common difference is 6, then its first term is
The first and last terms of an A.P. are 1 and 11. If the sum of its terms is 36, then the number of terms will be
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 second, third and sixth terms of an A.P. are consecutive terms of a G.P., write the common ratio of the G.P.
The product of three numbers in A.P. is 224, and the largest number is 7 times the smallest. Find the numbers
If 9 times the 9th term of an A.P. is equal to 13 times the 13th term, then the 22nd term of the A.P. is ______.
Let Sn denote the sum of the first n terms of an A.P. If S2n = 3Sn then S3n: Sn is equal to ______.
If n AM's are inserted between 1 and 31 and ratio of 7th and (n – 1)th A.M. is 5:9, then n equals ______.
If b2, a2, c2 are in A.P., then `1/(a + b), 1/(b + c), 1/(c + a)` will be in ______
