Advertisements
Advertisements
प्रश्न
Write the sum of first n odd natural numbers.
Advertisements
उत्तर
We need to find the sum of 1, 3, 5, 7... upto n terms.
Here, a = 1, d = 2
We know:
\[S_n = \frac{n}{2}\left\{ 2a + \left( n - 1 \right)d \right\}\]
\[ = \frac{n}{2}\left\{ 2 \times 1 + \left( n - 1 \right)2 \right\}\]
\[ = n^2\]
Therefore, the sum of the first n odd numbers is \[n^2\] .
APPEARS IN
संबंधित प्रश्न
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.
If the sum of three numbers in A.P., is 24 and their product is 440, find the numbers.
Find the sum of integers from 1 to 100 that are divisible by 2 or 5.
A man deposited Rs 10000 in a bank at the rate of 5% simple interest annually. Find the amount in 15th year since he deposited the amount and also calculate the total amount after 20 years.
Let < an > be a sequence. Write the first five term in the following:
a1 = 1, an = an − 1 + 2, n ≥ 2
The nth term of a sequence is given by an = 2n2 + n + 1. Show that it is not an A.P.
If the sequence < an > is an A.P., show that am +n +am − n = 2am.
Is 68 a term of the A.P. 7, 10, 13, ...?
The 10th and 18th terms of an A.P. are 41 and 73 respectively. Find 26th term.
Find the 12th term from the following arithmetic progression:
3, 8, 13, ..., 253
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.
\[\text { If } \theta_1 , \theta_2 , \theta_3 , . . . , \theta_n \text { are in AP, whose common difference is d, then show that }\]
\[\sec \theta_1 \sec \theta_2 + \sec \theta_2 \sec \theta_3 + . . . + \sec \theta_{n - 1} \sec \theta_n = \frac{\tan \theta_n - \tan \theta_1}{\sin d} \left[ NCERT \hspace{0.167em} EXEMPLAR \right]\]
Three numbers are in A.P. If the sum of these numbers be 27 and the product 648, find the numbers.
The sum of three numbers in A.P. is 12 and the sum of their cubes is 288. Find the numbers.
Find the sum of the following arithmetic progression :
1, 3, 5, 7, ... to 12 terms
Find the sum of the following arithmetic progression :
3, 9/2, 6, 15/2, ... to 25 terms
Find the sum of the following serie:
2 + 5 + 8 + ... + 182
Find the sum of first n odd natural numbers.
Find the sum of the series:
3 + 5 + 7 + 6 + 9 + 12 + 9 + 13 + 17 + ... to 3n terms.
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 odd integers from 1 to 2001.
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.
If x, y, z are in A.P. and A1 is the A.M. of x and y and A2 is the A.M. of y and z, then prove that the A.M. of A1 and A2 is y.
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 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?
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.
If log 2, log (2x − 1) and log (2x + 3) are in A.P., write the value of x.
Write the value of n for which n th terms of the A.P.s 3, 10, 17, ... and 63, 65, 67, .... are equal.
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
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
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 the first, second and last term of an A.P are a, b and 2a respectively, then its sum is
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
The product of three numbers in A.P. is 224, and the largest number is 7 times the smallest. Find the numbers
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 the sum of n terms of an A.P. is given by Sn = 3n + 2n2, then the common difference of the A.P. is ______.
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 ______.
If n AM's are inserted between 1 and 31 and ratio of 7th and (n – 1)th A.M. is 5:9, then n equals ______.
