Advertisements
Advertisements
प्रश्न
Find the sum of the following arithmetic progressions:
`(x - y)/(x + y),(3x - 2y)/(x + y), (5x - 3y)/(x + y)`, .....to n terms
Advertisements
उत्तर
`(x - y)/(x + y),(3x - 2y)/(x + y), (5x - 3y)/(x + y)`, .....to n terms
Number of terms (n) = n
Number of terms (n) = n = `((x - y)/(x + y))`
Common difference of the A.P. (d) = `a_2 - a_1`
`= ((3x - 2)/(x + y)) - (x - y)/(x + y)`
`= ((3x - 2y) - (x - y))/(x +y)`
`= (3x - 2y - x + y)/(x + y)`
`= (2x - y)/(x - y)`
So using the formula we get
`S_n = n/2[2((x - y)/(x + y)) + (n - 1)((2x - y )/(x + y))]`
`= (n/2) [((2x - 2y)/(x + y)) + (n(2x - y)- 2x + y)/(x + y)]`
`= (n/2)[(2x -2y)/(x + y) + (((n (2x - y) - 2x + y))/(x + y))]`
Now, on further solving the above equation we get,
`= (n/2)((2x - 2y + n(2x - y) - 2x + y)/(x + y))`
`= (n/2) ((n(2x - y) - y)/(x + y))`
Therefore, the sum of first n terms for the given A.P. is `(n/2) ((n(2x - y) - y)/(x + y))`
APPEARS IN
संबंधित प्रश्न
The houses in a row numbered consecutively from 1 to 49. Show that there exists a value of x such that sum of numbers of houses preceding the house numbered x is equal to sum of the numbers of houses following x.
How many terms of the A.P. 65, 60, 55, .... be taken so that their sum is zero?
If the mth term of an A.P. is 1/n and the nth term is 1/m, show that the sum of mn terms is (mn + 1)
Find the sum of the odd numbers between 0 and 50.
A small terrace at a football field comprises 15 steps, each of which is 50 m long and built of solid concrete. Each step has a rise of `1/4` m and a tread of `1/2` m (See figure). Calculate the total volume of concrete required to build the terrace.
[Hint: Volume of concrete required to build the first step = `1/4 xx 1/2 xx 50 m^3`]
If 18, a, (b - 3) are in AP, then find the value of (2a – b)
First term and the common differences of an A.P. are 6 and 3 respectively; find S27.
Solution: First term = a = 6, common difference = d = 3, S27 = ?
Sn = `"n"/2 [square + ("n" - 1)"d"]` - Formula
Sn = `27/2 [12 + (27 - 1)square]`
= `27/2 xx square`
= 27 × 45
S27 = `square`
There are 25 rows of seats in an auditorium. The first row is of 20 seats, the second of 22 seats, the third of 24 seats, and so on. How many chairs are there in the 21st row ?
The 19th term of an A.P. is equal to three times its sixth term. If its 9th term is 19, find the A.P.
If Sn denotes the sum of first n terms of an A.P., prove that S12 = 3(S8 − S4).
Write 5th term from the end of the A.P. 3, 5, 7, 9, ..., 201.
If the sum of first n even natural numbers is equal to k times the sum of first n odd natural numbers, then k =
The sum of the first three terms of an Arithmetic Progression (A.P.) is 42 and the product of the first and third term is 52. Find the first term and the common difference.
The sum of the first 2n terms of the AP: 2, 5, 8, …. is equal to sum of the first n terms of the AP: 57, 59, 61, … then n is equal to ______.
The sum of all odd integers between 2 and 100 divisible by 3 is ______.
In an A.P., if Sn = 3n2 + 5n and ak = 164, find the value of k.
Find the sum of those integers between 1 and 500 which are multiples of 2 as well as of 5.
Find the sum of first 16 terms of the A.P. whose nth term is given by an = 5n – 3.
Which term of the Arithmetic Progression (A.P.) 15, 30, 45, 60...... is 300?
Hence find the sum of all the terms of the Arithmetic Progression (A.P.)
In a ‘Mahila Bachat Gat’, Kavita invested from the first day of month ₹ 20 on first day, ₹ 40 on second day and ₹ 60 on third day. If she saves like this, then what would be her total savings in the month of February 2020?
