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
विकल्प
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
Find the sum of odd integers from 1 to 2001.
Find the sum of all natural numbers lying between 100 and 1000, which are multiples of 5.
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)
A man starts repaying a loan as first installment of Rs. 100. If he increases the installment by Rs 5 every month, what amount he will pay in the 30th installment?
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 all two digit numbers which when divided by 4, yields 1 as remainder.
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?
Which term of the A.P. 4, 9, 14, ... is 254?
Find the second term and nth term of an A.P. whose 6th term is 12 and the 8th term is 22.
Three numbers are in A.P. If the sum of these numbers be 27 and the product 648, find the numbers.
Find the four numbers in A.P., whose sum is 50 and in which the greatest number is 4 times the least.
The angles of a quadrilateral are in A.P. whose common difference is 10°. Find the angles.
Find the sum of the following arithmetic progression :
50, 46, 42, ... to 10 terms
Find the sum of the following serie:
2 + 5 + 8 + ... + 182
Find the sum of all those integers between 100 and 800 each of which on division by 16 leaves the remainder 7.
Solve:
1 + 4 + 7 + 10 + ... + x = 590.
The number of terms of an A.P. is even; the sum of odd terms is 24, of the even terms is 30, and the last term exceeds the first by \[10 \frac{1}{2}\] , find the number of terms and the series.
How many terms of the A.P. −6, \[- \frac{11}{2}\], −5, ... are needed to give the sum −25?
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).
If the sum of n terms of an A.P. is nP + \[\frac{1}{2}\] n (n − 1) Q, where P and Q are constants, find the common difference.
If \[\frac{1}{a}, \frac{1}{b}, \frac{1}{c}\] are in A.P., prove that:
a (b +c), b (c + a), c (a +b) are in A.P.
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.
If a, b, c is in A.P., prove that:
a2 + c2 + 4ac = 2 (ab + bc + ca)
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.
A man is employed to count Rs 10710. He counts at the rate of Rs 180 per minute for half an hour. After this he counts at the rate of Rs 3 less every minute than the preceding minute. Find the time taken by him to count the entire amount.
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 \[\frac{3 + 5 + 7 + . . . + \text { upto n terms }}{5 + 8 + 11 + . . . . \text { upto 10 terms }}\] 7, then find the value of n.
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
Mark the correct alternative in the following question:
\[\text { If in an A . P } . S_n = n^2 q \text { and } S_m = m^2 q, \text { where } S_r \text{ denotes the sum of r terms of the A . P . , then }S_q \text { equals }\]
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 for an arithmetic progression, 9 times nineth term is equal to 13 times thirteenth term, then value of twenty second term is ____________.
If there are (2n + 1) terms in an A.P., then prove that the ratio of the sum of odd terms and the sum of even terms is (n + 1) : n
The product of three numbers in A.P. is 224, and the largest number is 7 times the smallest. Find the numbers
Any term of an A.P. (except first) is equal to half the sum of terms which are equidistant from it.
