Advertisements
Advertisements
प्रश्न
Find the sum of the following arithmetic progression :
(x − y)2, (x2 + y2), (x + y)2, ... to n terms
उत्तर
(x − y)2, (x2 + y2), (x + y)2 ... to n terms
\[\text { We have }: \]
\[ a = {(x -y)}^2 , d = \left( x^2 + y^2 - {(x - y)}^2 \right) = 2xy\]
\[ S_n = \frac{n}{2}\left[ 2a + (n - 1)d \right]\]
\[ = \frac{n}{2}\left[ 2 {(x - y)}^2 + (n - 1)(2xy) \right]\]
\[ = \frac{n}{2} \times 2\left[ {(x -y)}^2 + (n - 1)(xy) \right]\]
\[ = n\left[ {(x - y)}^2 + (n - 1)(xy) \right]\]
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.
A manufacturer reckons that the value of a machine, which costs him Rs 15625, will depreciate each year by 20%. Find the estimated value at the end of 5 years.
The Fibonacci sequence is defined by a1 = 1 = a2, an = an − 1 + an − 2 for n > 2
Find `(""^an +1)/(""^an")` for n = 1, 2, 3, 4, 5.
Which term of the sequence 12 + 8i, 11 + 6i, 10 + 4i, ... is purely real ?
How many terms are there in the A.P.\[- 1, - \frac{5}{6}, -\frac{2}{3}, - \frac{1}{2}, . . . , \frac{10}{3}?\]
The 6th and 17th terms of an A.P. are 19 and 41 respectively, find the 40th term.
If 10 times the 10th term of an A.P. is equal to 15 times the 15th term, show that 25th term of the A.P. is zero.
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.
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.
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 :
3, 9/2, 6, 15/2, ... to 25 terms
Find the sum of the following arithmetic progression :
a + b, a − b, a − 3b, ... to 22 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 all those integers between 100 and 800 each of which on division by 16 leaves the remainder 7.
Solve:
25 + 22 + 19 + 16 + ... + x = 115
The first term of an A.P. is 2 and the last term is 50. The sum of all these terms is 442. Find the common difference.
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.
If a, b, c is in A.P., then show that:
bc − a2, ca − b2, ab − c2 are 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 starts repaying a loan as first instalment of Rs 100 = 00. If he increases the instalments by Rs 5 every month, what amount he will pay in the 30th instalment?
A man is employed to count Rs 10710. He counts at the rate of Rs 180 per minute for half an hour. After this he counts at the rate of Rs 3 less every minute than the preceding minute. Find the time taken by him to count the entire amount.
A man accepts a position with an initial salary of ₹5200 per month. It is understood that he will receive an automatic increase of ₹320 in the very next month and each month thereafter.
(i) Find his salary for the tenth month.
(ii) What is his total earnings during the first year?
In a cricket team tournament 16 teams participated. A sum of ₹8000 is to be awarded among themselves as prize money. If the last place team is awarded ₹275 in prize money and the award increases by the same amount for successive finishing places, then how much amount will the first place team receive?
Write the common difference of an A.P. the sum of whose first n terms is
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 7th and 13th terms of an A.P. be 34 and 64 respectively, then its 18th term is
Sum of all two digit numbers which when divided by 4 yield unity as remainder is
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
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:
Let Sn denote the sum of first n terms of an A.P. If S2n = 3Sn, then S3n : Sn is equal to
The pth term of an A.P. is a and qth term is b. Prove that the sum of its (p + q) terms is `(p + q)/2[a + b + (a - b)/(p - q)]`.
Any term of an A.P. (except first) is equal to half the sum of terms which are equidistant from it.
