Advertisements
Advertisements
प्रश्न
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
पर्याय
4
6
8
10
Advertisements
उत्तर
\[As, S_{2n} = 3 S_n \]
\[ \Rightarrow \frac{2n}{2}\left[ 2a + \left( 2n - 1 \right)d \right] = \frac{3n}{2}\left[ 2a + \left( n - 1 \right)d \right]\]
\[ \Rightarrow 2\left[ 2a + \left( 2n - 1 \right)d \right] = 3\left[ 2a + \left( n - 1 \right)d \right]\]
\[ \Rightarrow 4a + 2\left( 2n - 1 \right)d = 6a + 3\left( n - 1 \right)d\]
\[ \Rightarrow 4a + 4nd - 2d = 6a + 3nd - 3d\]
\[ \Rightarrow 6a - 4a + 3nd - 3d - 4nd + 2d = 0\]
\[ \Rightarrow 2a - nd - d = 0\]
\[ \Rightarrow 2a - \left( n + 1 \right)d = 0\]
\[ \Rightarrow 2a = \left( n + 1 \right)d . . . . . \left( i \right)\]
\[\text { Now }, \]
\[\frac{S_{3n}}{S_n} = \frac{\frac{3n}{2}\left[ 2a + \left( 3n - 1 \right)d \right]}{\frac{n}{2}\left[ 2a + \left( n - 1 \right)d \right]}\]
\[ = \frac{3\left[ \left( n + 1 \right)d + \left( 3n - 1 \right)d \right]}{\left[ \left( n + 1 \right)d + \left( n - 1 \right)d \right]} \left[ \text { Using } \left( i \right) \right]\]
\[ = \frac{3\left[ nd + d + 3nd - d \right]}{\left[ nd + d + nd - d \right]}\]
\[ = \frac{3\left[ 4nd \right]}{\left[ 2nd \right]}\]
\[ = 3 \times 2\]
\[ = 6\]
Hence, the correct alternative is option (b).
APPEARS IN
संबंधित प्रश्न
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 the sum of three numbers in A.P., is 24 and their product is 440, find the numbers.
Find the sum of all two digit numbers which when divided by 4, yields 1 as remainder.
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?
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, ...
The nth term of a sequence is given by an = 2n + 7. Show that it is an A.P. Also, find its 7th term.
The nth term of a sequence is given by an = 2n2 + n + 1. Show that it is not an A.P.
Which term of the A.P. 3, 8, 13, ... is 248?
The 10th and 18th terms of an A.P. are 41 and 73 respectively. Find 26th term.
If the nth term of the A.P. 9, 7, 5, ... is same as the nth term of the A.P. 15, 12, 9, ... find n.
The sum of three numbers in A.P. is 12 and the sum of their cubes is 288. Find the numbers.
Find the sum of the following arithmetic progression :
41, 36, 31, ... to 12 terms
Find the sum of the following serie:
2 + 5 + 8 + ... + 182
Find the sum of the following serie:
(a − b)2 + (a2 + b2) + (a + b)2 + ... + [(a + b)2 + 6ab]
Find the r th term of an A.P., the sum of whose first n terms is 3n2 + 2n.
If Sn = n2 p and Sm = m2 p, m ≠ n, in an A.P., prove that Sp = p3.
If 12th term of an A.P. is −13 and the sum of the first four terms is 24, what is the sum of first 10 terms?
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).
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 a, b, c is in A.P., then show that:
a2 (b + c), b2 (c + a), c2 (a + b) are also in A.P.
If a, b, c is in A.P., then show that:
bc − a2, ca − b2, ab − c2 are in A.P.
If a, b, c is in A.P., prove that:
a2 + c2 + 4ac = 2 (ab + bc + ca)
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. whose nth term is xn + y.
Write the value of n for which n th terms of the A.P.s 3, 10, 17, ... and 63, 65, 67, .... are equal.
Sum of all two digit numbers which when divided by 4 yield unity as remainder is
In n A.M.'s are introduced between 3 and 17 such that the ratio of the last mean to the first mean is 3 : 1, then the value of n is
In the arithmetic progression whose common difference is non-zero, the sum of first 3 n terms is equal to the sum of next n terms. Then the ratio of the sum of the first 2 n terms to the next 2 nterms is
The first and last term of an A.P. are a and l respectively. If S is the sum of all the terms of the A.P. and the common difference is given by \[\frac{l^2 - a^2}{k - (l + a)}\] , then k =
Mark the correct alternative in the following question:
If in an A.P., the pth term is q and (p + q)th term is zero, then the qth term is
If a, b, c are in A.P. and x, y, z are in G.P., then the value of xb − c yc − a za − b is
The product of three numbers in A.P. is 224, and the largest number is 7 times the smallest. Find the numbers
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 ______.
If 100 times the 100th term of an A.P. with non zero common difference equals the 50 times its 50th term, then the 150th term of this A.P. is ______.
The number of terms in an A.P. is even; the sum of the odd terms in lt is 24 and that the even terms is 30. If the last term exceeds the first term by `10 1/2`, then the number of terms in the A.P. is ______.
The fourth term of an A.P. is three times of the first term and the seventh term exceeds the twice of the third term by one, then the common difference of the progression is ______.
