Advertisements
Advertisements
प्रश्न
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
विकल्प
\[\frac{1}{2} p^3\]
mn p
P3
(m + n) p2
Advertisements
उत्तर
p3
Given:
\[S_n = n^2 p\]
\[ \Rightarrow \frac{n}{2}\left\{ 2a + \left( n - 1 \right)d \right\} = n^2 p\]
\[ \Rightarrow 2a + \left( n - 1 \right)d = 2np\]
\[ \Rightarrow 2a = 2np - \left( n - 1 \right)d . . . . . \left( 1 \right)\]
\[ S_m = m^2 p\]
\[ \Rightarrow \frac{m}{2}\left\{ 2a + \left( m - 1 \right)d \right\} = m^2 p\]
\[ \Rightarrow 2a + \left( m - 1 \right)d = 2mp\]
\[ \Rightarrow 2a = 2mp - \left( m - 1 \right)d . . . . . \left( 2 \right)\]
From
\[\left( 1 \right) \text { and } \left( 2 \right)\] , we have:
\[2np - \left( n - 1 \right)d = 2mp - \left( m - 1 \right)d\]
\[ \Rightarrow 2p\left( n - m \right) = d\left( n - 1 - m + 1 \right)\]
\[ \Rightarrow 2p = d\]
Substituting d = 2p in equation \[\left( 1 \right)\], we get:
a = p
Sum of p terms of the A.P. is given by:
\[\frac{p}{2}\left\{ 2a + \left( p - 1 \right)d \right\}\]
\[ = \frac{p}{2}\left\{ 2p + \left( p - 1 \right)2p \right\} \]
\[ = p^3\]
संबंधित प्रश्न
How many terms of the A.P. -6 , `-11/2` , -5... are needed to give the sum –25?
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`
Let the sum of n, 2n, 3n terms of an A.P. be S1, S2 and S3, respectively, show that S3 = 3 (S2– S1)
A farmer buys a used tractor for Rs 12000. He pays Rs 6000 cash and agrees to pay the balance in annual installments of Rs 500 plus 12% interest on the unpaid amount. How much will be the tractor cost him?
Find:
18th term of the A.P.
\[\sqrt{2}, 3\sqrt{2}, 5\sqrt{2},\]
Which term of the A.P. 3, 8, 13, ... is 248?
Which term of the A.P. 84, 80, 76,... is 0?
Is 302 a term of the A.P. 3, 8, 13, ...?
Which term of the sequence 12 + 8i, 11 + 6i, 10 + 4i, ... is purely real ?
How many terms are there in the A.P.\[- 1, - \frac{5}{6}, -\frac{2}{3}, - \frac{1}{2}, . . . , \frac{10}{3}?\]
Find the 12th term from the following arithmetic progression:
1, 4, 7, 10, ..., 88
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.
The first and the last terms of an A.P. are a and l respectively. Show that the sum of nthterm from the beginning and nth term from the end is a + l.
Find the sum of the following arithmetic progression :
50, 46, 42, ... to 10 terms
Find the sum of the following arithmetic progression :
(x − y)2, (x2 + y2), (x + y)2, ... to n terms
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.
Solve:
1 + 4 + 7 + 10 + ... + x = 590.
If Sn = n2 p and Sm = m2 p, m ≠ n, in an A.P., prove that Sp = p3.
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.
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 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:
a2 (b + c), b2 (c + a), c2 (a + b) are also in A.P.
Show that x2 + xy + y2, z2 + zx + x2 and y2 + yz + z2 are consecutive terms of an A.P., if x, y and z are in 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?
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.
Write the sum of first n even natural numbers.
If 7th and 13th terms of an A.P. be 34 and 64 respectively, then its 18th term is
In n A.M.'s are introduced between 3 and 17 such that the ratio of the last mean to the first mean is 3 : 1, then the value of n is
If the first, second and last term of an A.P are a, b and 2a respectively, then its sum is
Mark the correct alternative in the following question:
The 10th common term between the A.P.s 3, 7, 11, 15, ... and 1, 6, 11, 16, ... is
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
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 the sum of m terms of an A.P. is equal to the sum of either the next n terms or the next p terms, then prove that `(m + n) (1/m - 1/p) = (m + p) (1/m - 1/n)`
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 ______.
If the first term of an A.P. is 3 and the sum of its first 25 terms is equal to the sum of its next 15 terms, then the common difference of this A.P. is ______.
