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
Find the sum of odd integers from 1 to 2001.
Find the sum of all natural numbers lying between 100 and 1000, which are multiples of 5.
If the sum of a certain number of terms of the A.P. 25, 22, 19, … is 116. Find the last term
If the sum of n terms of an A.P. is (pn + qn2), where p and q are constants, 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
The ratio of the sums of m and n terms of an A.P. is m2: n2. Show that the ratio of mth and nthterm is (2m – 1): (2n – 1)
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.
The pth, qth and rth terms of an A.P. are a, b, c respectively. Show that (q – r )a + (r – p )b + (p – q )c = 0
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.
If the sequence < an > is an A.P., show that am +n +am − n = 2am.
How many terms are there in the A.P.\[- 1, - \frac{5}{6}, -\frac{2}{3}, - \frac{1}{2}, . . . , \frac{10}{3}?\]
Find the 12th term from the following arithmetic progression:
3, 8, 13, ..., 253
Find the second term and nth term of an A.P. whose 6th term is 12 and the 8th term is 22.
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 sum of three terms of an A.P. is 21 and the product of the first and the third terms exceeds the second term by 6, find three terms.
Find the four numbers in A.P., whose sum is 50 and in which the greatest number is 4 times the least.
Find the sum of the following arithmetic progression :
50, 46, 42, ... to 10 terms
Find the sum of the following arithmetic progression :
41, 36, 31, ... to 12 terms
Find the sum of the following arithmetic progression :
a + b, a − b, a − 3b, ... to 22 terms
Find the sum of all integers between 50 and 500 which are divisible by 7.
The number of terms of an A.P. is even; the sum of odd terms is 24, of the even terms is 30, and the last term exceeds the first by \[10 \frac{1}{2}\] , find the number of terms and the series.
If the 5th and 12th terms of an A.P. are 30 and 65 respectively, what is the sum of first 20 terms?
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 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.
If \[\frac{b + c}{a}, \frac{c + a}{b}, \frac{a + b}{c}\] are in A.P., prove that:
\[\frac{1}{a}, \frac{1}{b}, \frac{1}{c}\] are in 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.
In a potato race 20 potatoes are placed in a line at intervals of 4 meters with the first potato 24 metres from the starting point. A contestant is required to bring the potatoes back to the starting place one at a time. How far would he run in bringing back all the potatoes?
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?
A man saved ₹66000 in 20 years. In each succeeding year after the first year he saved ₹200 more than what he saved in the previous year. How much did he save in the first year?
Write the common difference of an A.P. the sum of whose first n terms is
If the sum of n terms of an A.P. be 3 n2 − n and its common difference is 6, then its first term is
If the sum of n terms of an A.P. is 2 n2 + 5 n, then its nth term 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, 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 \[\frac{S_1}{S_2}\] =
The first three of four given numbers are in G.P. and their last three are in A.P. with common difference 6. If first and fourth numbers are equal, then the first number is
The first term of an A.P. is a, the second term is b and the last term is c. Show that the sum of the A.P. is `((b + c - 2a)(c + a))/(2(b - a))`.
Find the sum of first 24 terms of the A.P. a1, a2, a3, ... if it is known that a1 + a5 + a10 + a15 + a20 + a24 = 225.
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 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 ______.
