Advertisements
Advertisements
प्रश्न
If Sn denotes the sum of first n terms of an A.P. < an > such that
पर्याय
\[\frac{2 m + 1}{2 n + 1}\]
\[\frac{2 m - 1}{2 n - 1}\]
\[\frac{m - 1}{n - 1}\]
\[\frac{m + 1}{n + 1}\]
Advertisements
उत्तर
\[\frac{2 m - 1}{2 n - 1}\]
\[\frac{S_m}{S_n} = \frac{m^2}{n^2}\]
\[ \Rightarrow \frac{\frac{m}{2}\left\{ 2a + \left( m - 1 \right)d \right\}}{\frac{n}{2}\left\{ 2a + \left( n - 1 \right)d \right\}} = \frac{m^2}{n^2}\]
\[ \Rightarrow \frac{\left\{ 2a + \left( m - 1 \right)d \right\}}{\left\{ 2a + \left( n - 1 \right)d \right\}} = \frac{m}{n} \]
\[ \Rightarrow 2an + ndm - nd = 2am + nmd - md\]
\[ \Rightarrow 2an - 2am - nd + md = 0\]
\[ \Rightarrow 2a\left( n - m \right) - d\left( n - m \right) = 0\]
\[ \Rightarrow 2a\left( n - m \right) = d\left( n - m \right)\]
\[ \Rightarrow d = 2a . . . . . \left( 1 \right)\]
\[\text { Ratio of } \frac{a_m}{a_n} = \frac{a + \left( m - 1 \right)d}{a + \left( n - 1 \right)d}\]
\[ \Rightarrow \frac{a_m}{a_n} = \frac{a + \left( m - \right)2a}{a + \left( n - \right)2a}\text { From } (1)\]
\[ = \frac{a + 2am - 2a}{a + 2an - 2a} \]
\[ = \frac{2am - a}{2an - a}\]
\[ = \frac{2m - 1}{2n - 1}\]
संबंधित प्रश्न
Find the sum of odd integers from 1 to 2001.
Sum of the first p, q and r terms of an A.P. are a, b and c, respectively.
Prove that `a/p (q - r) + b/q (r- p) + c/r (p - q) = 0`
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)
if `(a^n + b^n)/(a^(n-1) + b^(n-1))` is the A.M. between a and b, then find the value of n.
Let the sum of n, 2n, 3n terms of an A.P. be S1, S2 and S3, respectively, show that S3 = 3 (S2– S1)
Find:
nth term of the A.P. 13, 8, 3, −2, ...
How many numbers of two digit are divisible by 3?
How many numbers are there between 1 and 1000 which when divided by 7 leave remainder 4?
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 serie:
101 + 99 + 97 + ... + 47
Find the sum of the following serie:
(a − b)2 + (a2 + b2) + (a + b)2 + ... + [(a + b)2 + 6ab]
Find the sum of first n odd natural numbers.
Find the sum of all integers between 50 and 500 which are divisible by 7.
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.
If 12th term of an A.P. is −13 and the sum of the first four terms is 24, what is the sum of first 10 terms?
Find the sum of n terms of the A.P. whose kth terms is 5k + 1.
If the sum of a certain number of terms of the AP 25, 22, 19, ... is 116. Find the last term.
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 \[\frac{b + c}{a}, \frac{c + a}{b}, \frac{a + b}{c}\] are in A.P., prove that:
\[\frac{1}{a}, \frac{1}{b}, \frac{1}{c}\] are in A.P.
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?
If the sums of n terms of two arithmetic progressions are in the ratio 2n + 5 : 3n + 4, then write the ratio of their m th terms.
Write the sum of first n odd natural numbers.
Write the sum of first n even natural numbers.
If the sums of n terms of two AP.'s are in the ratio (3n + 2) : (2n + 3), then find the ratio of their 12th terms.
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
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 for an arithmetic progression, 9 times nineth term is equal to 13 times thirteenth term, then value of twenty second term is ____________.
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
If b2, a2, c2 are in A.P., then `1/(a + b), 1/(b + c), 1/(c + a)` will be in ______
The internal angles of a convex polygon are in A.P. The smallest angle is 120° and the common difference is 5°. The number to sides of the polygon is ______.
What is the general form of an A.P. with first term \(a\) and common difference \(d\)?
