Advertisements
Advertisements
Question
Let Sn denote the sum of the first n terms of an A.P. If S2n = 3Sn then S3n: Sn is equal to ______.
Options
4
6
8
10
Advertisements
Solution
Let Sn denote the sum of the first n terms of an A.P. If S2n = 3Sn then S3n: Sn is equal to 6.
Explanation:
Sn = `n/2 [2a + (n - 1)d]`
∴ S2n = `(2n)/2 [2a + (2n - 1)d]`
3Sn = `(3n)/2 [2a + (3n - 1)d]`
We have S2n = 3 · Sn
⇒ `(2n)/2 [2a + (2n - 1)d] = 3 * n/2 [2a + (n - 1)d]`
⇒ 2[2a + (2n – 1)d] = 3[2a + (n – 1)d]
⇒ 4a + (4n – 2)d = 6a + (3n – 3)d
⇒ 6a + (3n – 3)d – 4a – (4n – 2)d = 0
⇒ 2a + (3n – 3 – 4n + 2)d = 0
⇒ 2a + (– n – 1)d = 0
⇒ 2a – (n + 1)d = 0
⇒ 2a = (n + 1)d ....(i)
Now S3n: Sn = `(3n)/2 [2a + (3n - 1)d] : n/2 [2a + (n - 1)d]`
= `((3n)/2 [2a + (3n - 1)d])/(n/2 [2a + (n - 1)d])`
= `(3[2a + (3n - 1)d])/(2a + (n - 1)d)`
= `(3[(n + 1)d + (3n - 1)d])/((n + 1)d + (n - 1)d)`
= `(3d[n + 1 + 3n - 1])/(d(n + 1 + n - 1))`
= `(3[4n])/(2n)`
= 6
APPEARS IN
RELATED QUESTIONS
If the sum of n terms of an A.P. is (pn + qn2), where p and q are constants, find the common difference.
Insert five numbers between 8 and 26 such that the resulting sequence is an A.P.
Show that the sum of (m + n)th and (m – n)th terms of an A.P. is equal to twice the mth term.
Find the sum of integers from 1 to 100 that are divisible by 2 or 5.
if `a(1/b + 1/c), b(1/c+1/a), c(1/a+1/b)` are in A.P., prove that a, b, c are in A.P.
A farmer buys a used tractor for Rs 12000. He pays Rs 6000 cash and agrees to pay the balance in annual installments of Rs 500 plus 12% interest on the unpaid amount. How much will be the tractor cost him?
Which term of the sequence 12 + 8i, 11 + 6i, 10 + 4i, ... is purely imaginary?
The 6th and 17th terms of an A.P. are 19 and 41 respectively, find the 40th term.
In a certain A.P. the 24th term is twice the 10th term. Prove that the 72nd term is twice the 34th term.
Find the 12th term from the following arithmetic progression:
3, 5, 7, 9, ... 201
The 4th term of an A.P. is three times the first and the 7th term exceeds twice the third term by 1. Find the first term and the common difference.
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.
The angles of a quadrilateral are in A.P. whose common difference is 10°. Find the angles.
Find the sum of the following serie:
101 + 99 + 97 + ... + 47
Find the sum of the following serie:
(a − b)2 + (a2 + b2) + (a + b)2 + ... + [(a + b)2 + 6ab]
Find the sum of all even integers between 101 and 999.
Find the sum of all integers between 100 and 550, which are divisible by 9.
The third term of an A.P. is 7 and the seventh term exceeds three times the third term by 2. Find the first term, the common difference and the sum of first 20 terms.
If Sn = n2 p and Sm = m2 p, m ≠ n, in an A.P., prove that Sp = p3.
If S1 be the sum of (2n + 1) terms of an A.P. and S2 be the sum of its odd terms, then prove that: S1 : S2 = (2n + 1) : (n + 1).
Find an A.P. in which the sum of any number of terms is always three times the squared number of these terms.
A man arranges to pay off a debt of Rs 3600 by 40 annual instalments which form an arithmetic series. When 30 of the instalments are paid, he dies leaving one-third of the debt unpaid, find the value of the first instalment.
There are 25 trees at equal distances of 5 metres in a line with a well, the distance of the well from the nearest tree being 10 metres. A gardener waters all the trees separately starting from the well and he returns to the well after watering each tree to get water for the next. Find the total distance the gardener will cover in order to water all the trees.
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.
If the sum of n terms of an AP is 2n2 + 3n, then write its nth term.
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 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 the sum of n terms of an A.P., is 3 n2 + 5 n then which of its terms is 164?
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, S1 is the sum of an arithmetic progression of 'n' odd number of terms and S2 the sum of the terms of the series in odd places, then \[\frac{S_1}{S_2}\] =
If second, third and sixth terms of an A.P. are consecutive terms of a G.P., write the common ratio of the G.P.
The product of three numbers in A.P. is 224, and the largest number is 7 times the smallest. Find the numbers
A man saved Rs 66000 in 20 years. In each succeeding year after the first year he saved Rs 200 more than what he saved in the previous year. How much did he save in the first year?
If in an A.P., Sn = qn2 and Sm = qm2, where Sr denotes the sum of r terms of the A.P., then Sq equals ______.
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 ______.
