Advertisements
Advertisements
प्रश्न
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
पर्याय
0
p − q
p + q
− (p + q)
Advertisements
उत्तर
− (p + q)
\[S_p = q\]
\[ \Rightarrow \frac{p}{2}\left\{ 2a + \left( p - 1 \right)d \right\} = q\]
\[ \Rightarrow 2ap + \left( p - 1 \right)pd = 2q . . . . . \left( 1 \right)\]
\[ S_q = p\]
\[ \Rightarrow \frac{q}{2}\left\{ 2a + \left( q - 1 \right)d \right\} = p\]
\[ \Rightarrow 2aq + \left( q - 1 \right)qd = 2p . . . . . \left( 2 \right)\]
\[\text { Multiplying equation } \left( 1 \right) \text { by q and equation } \left( 2 \right) \text { by p and then solving, we get }: \]
\[d = \frac{- 2\left( p + q \right)}{pq}\]
\[\text { Now }, S_{p + q} = \frac{\left( p + q \right)}{2}\left[ 2a + \left( p + q - 1 \right)d \right]\]
\[ = \frac{p}{2}\left[ 2a + \left( p - 1 \right)d + qd \right] + \frac{q}{2}\left[ 2a + \left( q - 1 \right)d + pd \right]\]
\[ = S_p + \frac{pqd}{2} + S_q + \frac{pqd}{2}\]
\[ = p + q + pqd\]
\[ = p + q - \frac{2\left( p + q \right)pq}{pq}\]
\[ = - (p + q)\]
APPEARS IN
संबंधित प्रश्न
In an A.P., if pth term is 1/q and qth term is 1/p, prove that the sum of first pq terms is 1/2 (pq + 1) where `p != q`
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.
Show that the sum of (m + n)th and (m – n)th terms of an A.P. is equal to twice the mth term.
If the sum of three numbers in A.P., is 24 and their product is 440, find the numbers.
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 ...
Show that the following sequence is an A.P. Also find the common difference and write 3 more terms in case.
−1, 1/4, 3/2, 11/4, ...
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.
Which term of the A.P. 84, 80, 76, ... is 0?
Is 68 a term of the A.P. 7, 10, 13, ...?
Is 302 a term of the A.P. 3, 8, 13, ...?
Which term of the sequence 12 + 8i, 11 + 6i, 10 + 4i, ... is purely imaginary?
The 6th and 17th terms of an A.P. are 19 and 41 respectively, find the 40th term.
If 10 times the 10th term of an A.P. is equal to 15 times the 15th term, show that 25th term of the A.P. is zero.
If (m + 1)th term of an A.P. is twice the (n + 1)th term, prove that (3m + 1)th term is twice the (m + n + 1)th term.
Find the r th term of an A.P., the sum of whose first n terms is 3n2 + 2n.
How many terms are there in the A.P. whose first and fifth terms are −14 and 2 respectively and the sum of the terms is 40?
The sum of first 7 terms of an A.P. is 10 and that of next 7 terms is 17. Find the progression.
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).
Find an A.P. in which the sum of any number of terms is always three times the squared number of these terms.
The sums of first n terms of two A.P.'s are in the ratio (7n + 2) : (n + 4). Find the ratio of their 5th terms.
If a, b, c is in A.P., then show that:
a2 (b + c), b2 (c + a), c2 (a + b) are also in A.P.
A manufacturer of radio sets produced 600 units in the third year and 700 units in the seventh year. Assuming that the product increases uniformly by a fixed number every year, find (i) the production in the first year (ii) the total product in 7 years and (iii) the product in the 10th year.
There are 25 trees at equal distances of 5 metres in a line with a well, the distance of the well from the nearest tree being 10 metres. A gardener waters all the trees separately starting from the well and he returns to the well after watering each tree to get water for the next. Find the total distance the gardener will cover in order to water all the trees.
A farmer buys a used tractor for Rs 12000. He pays Rs 6000 cash and agrees to pay the balance in annual instalments of Rs 500 plus 12% interest on the unpaid amount. How much the tractor cost him?
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?
A man saved ₹66000 in 20 years. In each succeeding year after the first year he saved ₹200 more than what he saved in the previous year. How much did he save in the first year?
If m th term of an A.P. is n and nth term is m, then write its pth term.
If Sn denotes the sum of first n terms of an A.P. < an > such that
Let Sn denote the sum of n terms of an A.P. whose first term is a. If the common difference d is given by d = Sn − k Sn − 1 + Sn − 2 , then k =
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
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
If a, b, c are in G.P. and a1/x = b1/y = c1/z, then xyz are in
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.
In an A.P. the pth term is q and the (p + q)th term is 0. Then the qth term is ______.
A man saved Rs 66000 in 20 years. In each succeeding year after the first year he saved Rs 200 more than what he saved in the previous year. How much did he save in the first year?
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 ______.
The sum of n terms of an AP is 3n2 + 5n. The number of term which equals 164 is ______.
