Advertisements
Advertisements
प्रश्न
If the sum of first n even natural numbers is equal to k times the sum of first n odd natural numbers, then k =
पर्याय
\[\frac{1}{n}\]
\[\frac{n - 1}{n}\]
\[\frac{n + 1}{2n}\]
\[\frac{n + 1}{n}\]
Advertisements
उत्तर
\[\frac{n + 1}{n}\]
Given:
Sum of the even natural numbers = k\[\times\] Sum of the odd natural numbers
\[\frac{n}{2}\left\{ 2a + \left( n - 1 \right)d \right\} = k \times \frac{n}{2}\left\{ 2a + \left( n - 1 \right)d \right\}\]
\[ \Rightarrow \left\{ 2 \times 2 + \left( n - 1 \right)2 \right\} = k \times \left\{ 2 \times 1 + \left( n - 1 \right)2 \right\}\]
\[ \Rightarrow \frac{4 + \left( n - 1 \right)2}{2 + \left( n - 1 \right)2} = k\]
\[ \Rightarrow \frac{n + 1}{n} = k\]
संबंधित प्रश्न
Find the sum of odd integers from 1 to 2001.
How many terms of the A.P. -6 , `-11/2` , -5... are needed to give the sum –25?
The sums of n terms of two arithmetic progressions are in the ratio 5n + 4: 9n + 6. Find the ratio of their 18th terms
If the sum of n terms of an A.P. is 3n2 + 5n and its mth term is 164, find the value of m.
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)
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?
If the nth term an of a sequence is given by an = n2 − n + 1, write down its first five terms.
Is 68 a term of the A.P. 7, 10, 13, ...?
The first term of an A.P. is 5, the common difference is 3 and the last term is 80; find the number of terms.
If 9th term of an A.P. is zero, prove that its 29th term is double the 19th term.
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.
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 :
\[\frac{x - y}{x + y}, \frac{3x - 2y}{x + y}, \frac{5x - 3y}{x + y}\], ... to n terms.
Find the sum of first n natural numbers.
Find the sum of first n odd natural numbers.
Find the sum of all those integers between 100 and 800 each of which on division by 16 leaves the remainder 7.
If Sn = n2 p and Sm = m2 p, m ≠ n, in an A.P., prove that Sp = p3.
Find the sum of odd integers from 1 to 2001.
The sums of n terms of two arithmetic progressions are in the ratio 5n + 4 : 9n + 6. Find the ratio of their 18th terms.
If a, b, c is in A.P., then show that:
b + c − a, c + a − b, a + b − c are in A.P.
A man saves Rs 32 during the first year. Rs 36 in the second year and in this way he increases his savings by Rs 4 every year. Find in what time his saving will be Rs 200.
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?
Write the sum of first n odd natural numbers.
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
If n arithmetic means are inserted between 1 and 31 such that the ratio of the first mean and nth mean is 3 : 29, then the value of n 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
If a, b, c, d are four distinct positive quantities in A.P., then show that bc > ad
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. What is his total earnings during 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 terms equidistant from the beginning and end in an A.P. is equal to ______.
If the ratio of the sum of n terms of two APs is 2n:(n + 1), then the ratio of their 8th terms is ______.
If 100 times the 100th term of an A.P. with non zero common difference equals the 50 times its 50th term, then the 150th term of this A.P. is ______.
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 \(d\)?
Which type of A.P. is represented by the sequence \(2, 5, 8, 11, \ldots\)?
