Advertisements
Advertisements
Question
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
Options
4
6
8
10
Advertisements
Solution
\[As, S_{2n} = 3 S_n \]
\[ \Rightarrow \frac{2n}{2}\left[ 2a + \left( 2n - 1 \right)d \right] = \frac{3n}{2}\left[ 2a + \left( n - 1 \right)d \right]\]
\[ \Rightarrow 2\left[ 2a + \left( 2n - 1 \right)d \right] = 3\left[ 2a + \left( n - 1 \right)d \right]\]
\[ \Rightarrow 4a + 2\left( 2n - 1 \right)d = 6a + 3\left( n - 1 \right)d\]
\[ \Rightarrow 4a + 4nd - 2d = 6a + 3nd - 3d\]
\[ \Rightarrow 6a - 4a + 3nd - 3d - 4nd + 2d = 0\]
\[ \Rightarrow 2a - nd - d = 0\]
\[ \Rightarrow 2a - \left( n + 1 \right)d = 0\]
\[ \Rightarrow 2a = \left( n + 1 \right)d . . . . . \left( i \right)\]
\[\text { Now }, \]
\[\frac{S_{3n}}{S_n} = \frac{\frac{3n}{2}\left[ 2a + \left( 3n - 1 \right)d \right]}{\frac{n}{2}\left[ 2a + \left( n - 1 \right)d \right]}\]
\[ = \frac{3\left[ \left( n + 1 \right)d + \left( 3n - 1 \right)d \right]}{\left[ \left( n + 1 \right)d + \left( n - 1 \right)d \right]} \left[ \text { Using } \left( i \right) \right]\]
\[ = \frac{3\left[ nd + d + 3nd - d \right]}{\left[ nd + d + nd - d \right]}\]
\[ = \frac{3\left[ 4nd \right]}{\left[ 2nd \right]}\]
\[ = 3 \times 2\]
\[ = 6\]
Hence, the correct alternative is option (b).
RELATED QUESTIONS
In an A.P, the first term is 2 and the sum of the first five terms is one-fourth of the next five terms. Show that 20th term is –112.
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.
Let < an > be a sequence. Write the first five term in the following:
a1 = 1 = a2, an = an − 1 + an − 2, n > 2
Let < an > be a sequence. Write the first five term in the following:
a1 = a2 = 2, an = an − 1 − 1, n > 2
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.
Is 302 a term of the A.P. 3, 8, 13, ...?
How many terms are there in the A.P. 7, 10, 13, ... 43 ?
The first term of an A.P. is 5, the common difference is 3 and the last term is 80; find the number of terms.
If 10 times the 10th term of an A.P. is equal to 15 times the 15th term, show that 25th term of the A.P. is zero.
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.
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 :
\[\frac{x - y}{x + y}, \frac{3x - 2y}{x + y}, \frac{5x - 3y}{x + y}\], ... to n terms.
Find the sum of all integers between 50 and 500 which are divisible by 7.
Find the sum of all integers between 100 and 550, which are divisible by 9.
Find the sum of the series:
3 + 5 + 7 + 6 + 9 + 12 + 9 + 13 + 17 + ... to 3n terms.
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 Sn = n2 p and Sm = m2 p, m ≠ n, in an A.P., prove that Sp = p3.
Find the sum of odd integers from 1 to 2001.
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 a, b, c is in A.P., then show that:
b + c − a, c + a − b, a + b − c are 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 \[a\left( \frac{1}{b} + \frac{1}{c} \right), b\left( \frac{1}{c} + \frac{1}{a} \right), c\left( \frac{1}{a} + \frac{1}{b} \right)\] are in A.P., prove that a, b, c are in A.P.
Insert five numbers between 8 and 26 such that the resulting sequence is an A.P.
A man saves Rs 32 during the first year. Rs 36 in the second year and in this way he increases his savings by Rs 4 every year. Find in what time his saving will be Rs 200.
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.
A farmer buys a used tractor for Rs 12000. He pays Rs 6000 cash and agrees to pay the balance in annual instalments of Rs 500 plus 12% interest on the unpaid amount. How much the tractor cost him?
The income of a person is Rs 300,000 in the first year and he receives an increase of Rs 10000 to his income per year for the next 19 years. Find the total amount, he received in 20 years.
If the sums of n terms of two arithmetic progressions are in the ratio 2n + 5 : 3n + 4, then write the ratio of their m th terms.
Mark the correct alternative in the following question:
The 10th common term between the A.P.s 3, 7, 11, 15, ... and 1, 6, 11, 16, ... is
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
Let 3, 6, 9, 12 ....... upto 78 terms and 5, 9, 13, 17 ...... upto 59 be two series. Then, the sum of the terms common to both the series is equal to ______.
If the ratio of the sum of n terms of two APs is 2n:(n + 1), then the ratio of their 8th terms is ______.
The sum of n terms of an AP is 3n2 + 5n. The number of term which equals 164 is ______.
If b2, a2, c2 are in A.P., then `1/(a + b), 1/(b + c), 1/(c + a)` will be in ______
