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
In an A.P., if pth term is 1/q and qth term is 1/p, prove that the sum of first pq terms is 1/2 (pq + 1) where `p != q`
If the sum of a certain number of terms of the A.P. 25, 22, 19, … is 116. Find the last term
Find the sum to n terms of the A.P., whose kth term is 5k + 1.
Find the sum of all numbers between 200 and 400 which are divisible by 7.
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.
Show that the following sequence is an A.P. Also find the common difference and write 3 more terms in case.
3, −1, −5, −9 ...
Show that the following sequence is an A.P. Also find the common difference and write 3 more terms in case.
−1, 1/4, 3/2, 11/4, ...
Show that the following sequence is an A.P. Also find the common difference and write 3 more terms in case.
\[\sqrt{2}, 3\sqrt{2}, 5\sqrt{2}, 7\sqrt{2}, . . .\]
Which term of the A.P. 3, 8, 13, ... is 248?
Which term of the A.P. 84, 80, 76, ... is 0?
Which term of the A.P. 4, 9, 14, ... is 254?
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.
Find the four numbers in A.P., whose sum is 50 and in which the greatest number is 4 times the least.
If the sum of three numbers in A.P. is 24 and their product is 440, 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 :
(x − y)2, (x2 + y2), (x + y)2, ... to n terms
Find the sum of the following serie:
(a − b)2 + (a2 + b2) + (a + b)2 + ... + [(a + b)2 + 6ab]
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.
If the sum of a certain number of terms of the AP 25, 22, 19, ... is 116. Find the last term.
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).
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 \[\frac{1}{a}, \frac{1}{b}, \frac{1}{c}\] are in A.P., prove that:
a (b +c), b (c + a), c (a +b) are in A.P.
If a2, b2, c2 are in A.P., prove that \[\frac{a}{b + c}, \frac{b}{c + a}, \frac{c}{a + b}\] are in A.P.
Insert five numbers between 8 and 26 such that the resulting sequence is an A.P.
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 piece of equipment cost a certain factory Rs 600,000. If it depreciates in value, 15% the first, 13.5% the next year, 12% the third year, and so on. What will be its value at the end of 10 years, all percentages applying to the original cost?
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.
Write the common difference of an A.P. the sum of whose first n terms is
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.
The first and last terms of an A.P. are 1 and 11. If the sum of its terms is 36, then the number of terms will be
If a1, a2, a3, .... an are in A.P. with common difference d, then the sum of the series sin d [sec a1 sec a2 + sec a2 sec a3 + .... + sec an − 1 sec an], is
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 − k Sn − 1 + Sn − 2 , then k =
If a, b, c are in G.P. and a1/x = b1/y = c1/z, then xyz are in
If there are (2n + 1) terms in an A.P., then prove that the ratio of the sum of odd terms and the sum of even terms is (n + 1) : n
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 ______.
