Advertisements
Advertisements
प्रश्न
If the ratio of sum of the first m and n terms of an AP is m2 : n2, show that the ratio of its mth and nth terms is (2m − 1) : (2n − 1) ?
Advertisements
उत्तर
Let the first term and the common difference of the AP be a and d, respectively.
Therefore,
Sum of the first m terms of the AP,
`"S"_"m" = "m"/2 [2"a" + ("m - 1")"d"]`
Sum of the first n terms of the AP,
`"S"_"n" = "n"/2[2"a" + ("n - 1")"d"]`
It is given that
`"S"_"m"/"S"_"n" = ("m"/2 [2"a" + ("m - 1")"d"])/("n"/2[2"a" + ("n - 1")"d"])`
`=> ([2"a" + ("m - 1")"d"])/([2"a" + ("n - 1")"d"]) = "m"/"n"`
⇒ 2an + mnd - nd = 2am + nmd - md
⇒ 2an - 2am = nd - md
⇒ 2a(n - m) = d(n - m)
⇒ 2a = d
Now,
`"T"_"m"/"T"_"n" = ("a" + ("m" - 1)"d")/("a" + ("n" - 1)"d")`
`=> "T"_"m"/"T"_"n" = ("a" + ("m - 1") xx 2"a")/("a" + ("n" - 1) xx 2"a")`
`=> "T"_"m"/"T"_"n" = (2"m" - 1)/("2n" - 1)`
Hence, the ratio of the mth term to the nth term is (2m − 1) : (2n − 1).
संबंधित प्रश्न
Find the 20th term from the last term of the A.P. 3, 8, 13, …, 253.
Find the sum given below:
34 + 32 + 30 + ... + 10
Find the sum:
25 + 28 + 31 + ... + 100
Find the sum of first 12 natural numbers each of which is a multiple of 7.
The 8th term of an AP is zero. Prove that its 38th term is triple its 18th term.
The sum of three consecutive terms of an AP is 21 and the sum of the squares of these terms is 165. Find these terms.
If the sum of first p terms of an AP is (ap2 + bp), find its common difference.
The sum of the first n terms of an AP is `((5n^2)/2 + (3n)/2)`. Find the nth term and the 20th term of this AP.
In an A.P. 19th term is 52 and 38th term is 128, find sum of first 56 terms.
Choose the correct alternative answer for the following question .
First four terms of an A.P. are ....., whose first term is –2 and common difference is –2.
Find the sum of n terms of the series \[\left( 4 - \frac{1}{n} \right) + \left( 4 - \frac{2}{n} \right) + \left( 4 - \frac{3}{n} \right) + ....\]
Write the value of a30 – a10 for the A.P. 4, 9, 14, 19,....
Let Sn denote the sum of n terms of an A.P. whose first term is a. If the common difference d is given by d = Sn − kSn−1 + Sn−2, then k =
If Sn denote the sum of the first n terms of an A.P. If S2n = 3Sn, then S3n : Sn is equal to
The sum of n terms of an A.P. is 3n2 + 5n, then 164 is its
How many terms of the series 18 + 15 + 12 + ........ when added together will give 45?
If the sum of the first four terms of an AP is 40 and that of the first 14 terms is 280. Find the sum of its first n terms.
Find the sum of all members from 50 to 250 which divisible by 6 and find t13.
The sum of the first three numbers in an Arithmetic Progression is 18. If the product of the first and the third term is 5 times the common difference, find the three numbers.
If the nth term of an AP is (2n + 1), then the sum of its first three terms is ______.
