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
Find the sum of odd integers from 1 to 2001.
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.
Show that the sum of (m + n)th and (m – n)th terms of an A.P. is equal to twice the mth term.
Find the sum of integers from 1 to 100 that are divisible by 2 or 5.
If the nth term an of a sequence is given by an = n2 − n + 1, write down its first five terms.
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.
Show that the following sequence is an A.P. Also find the common difference and write 3 more terms in case.
3, −1, −5, −9 ...
Which term of the A.P. 84, 80, 76, ... is 0?
Which term of the sequence 24, \[23\frac{1}{4,} 22\frac{1}{2,} 21\frac{3}{4}\]....... is the first negative term?
Find the 12th term from the following arithmetic progression:
3, 5, 7, 9, ... 201
An A.P. consists of 60 terms. If the first and the last terms be 7 and 125 respectively, find 32nd term.
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.
\[\text { If } \theta_1 , \theta_2 , \theta_3 , . . . , \theta_n \text { are in AP, whose common difference is d, then show that }\]
\[\sec \theta_1 \sec \theta_2 + \sec \theta_2 \sec \theta_3 + . . . + \sec \theta_{n - 1} \sec \theta_n = \frac{\tan \theta_n - \tan \theta_1}{\sin d} \left[ NCERT \hspace{0.167em} EXEMPLAR \right]\]
Three numbers are in A.P. If the sum of these numbers be 27 and the product 648, find the numbers.
Find the sum of the following arithmetic progression :
50, 46, 42, ... to 10 terms
Find the sum of first n natural numbers.
Find the sum of first n odd natural numbers.
Find the r th term of an A.P., the sum of whose first n terms is 3n2 + 2n.
The sum of first 7 terms of an A.P. is 10 and that of next 7 terms is 17. Find the progression.
If the sum of a certain number of terms of the AP 25, 22, 19, ... is 116. Find the last term.
Find an A.P. in which the sum of any number of terms is always three times the squared number of these terms.
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 a, b, c is in A.P., prove that:
a3 + c3 + 6abc = 8b3.
If \[a\left( \frac{1}{b} + \frac{1}{c} \right), b\left( \frac{1}{c} + \frac{1}{a} \right), c\left( \frac{1}{a} + \frac{1}{b} \right)\] are in A.P., prove that a, b, c are in A.P.
A piece of equipment cost a certain factory Rs 600,000. 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?
We know that the sum of the interior angles of a triangle is 180°. Show that the sums of the interior angles of polygons with 3, 4, 5, 6, ... sides form an arithmetic progression. Find the sum of the interior angles for a 21 sided polygon.
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 sum of first n odd natural numbers.
If the sum of n terms of an A.P. is 2 n2 + 5 n, then its nth term 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 =
Mark the correct alternative in the following question:
If in an A.P., the pth term is q and (p + q)th term is zero, then the qth term is
Write the quadratic equation the arithmetic and geometric means of whose roots are Aand G respectively.
If a, b, c are in A.P. and x, y, z are in G.P., then the value of xb − c yc − a za − b is
If for an arithmetic progression, 9 times nineth term is equal to 13 times thirteenth term, then value of twenty second term is ____________.
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.
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
Find the rth term of an A.P. sum of whose first n terms is 2n + 3n2
If in an A.P., Sn = qn2 and Sm = qm2, where Sr denotes the sum of r terms of the A.P., then Sq equals ______.
Any term of an A.P. (except first) is equal to half the sum of terms which are equidistant from it.
