Advertisements
Advertisements
Question
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 \[\frac{S_1}{S_2}\] =
Options
\[\frac{2n}{n + 1}\]
\[\frac{n}{n + 1}\]
\[\frac{n + 1}{2n}\]
\[\frac{n + 1}{n}\]
Advertisements
Solution
\[\frac{2n}{n + 1}\]
Let n be an odd number.
Given:
\[S_1 = \text { Sum of odd number of terms }\]
\[ = \frac{n}{2}\left\{ 2a + \left( n - 1 \right)d \right\} . . . . . \left( 1 \right)\]
\[\text { Since n is odd, the number of odd places } = \frac{n + 1}{2}\]
\[ S_2 = \text { Sum of the terms of a series in odd places }\]
\[ = \frac{\left( \frac{n + 1}{2} \right)}{2}\left\{ 2a + \left( \frac{n + 1}{2} - 1 \right)2d \right\}\]
\[ = \frac{n + 1}{4}\left\{ 2a + \left( n - 1 \right)d \right\} . . . . . \left( 2 \right)\]
From equations \[\left( 1 \right) \text { and } \left( 2 \right)\] ,we have:
\[\frac{S_1}{S_2} = \frac{\frac{n}{2}\left\{ 2a + \left( n - 1 \right)d \right\}}{\frac{n + 1}{4}\left\{ 2a + \left( n - 1 \right)d \right\}}\]
\[ = \frac{2n}{n + 1}\]
APPEARS IN
RELATED QUESTIONS
Find the sum to n terms of the A.P., whose kth term is 5k + 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
The sum of the first four terms of an A.P. is 56. The sum of the last four terms is 112. If its first term is 11, then find the number of terms.
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.
Show that the following sequence is an A.P. Also find the common difference and write 3 more terms in case.
9, 7, 5, 3, ...
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.
Find:
10th term of the A.P. 1, 4, 7, 10, ...
Which term of the sequence 24, \[23\frac{1}{4,} 22\frac{1}{2,} 21\frac{3}{4}\]....... is the first negative term?
If (m + 1)th term of an A.P. is twice the (n + 1)th term, prove that (3m + 1)th term is twice the (m + n + 1)th term.
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.
How many numbers of two digit are divisible by 3?
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 :
(x − y)2, (x2 + y2), (x + y)2, ... to n terms
Find the sum of all odd numbers between 100 and 200.
Find the sum of all integers between 50 and 500 which are divisible by 7.
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.
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.
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.
Shamshad Ali buys a scooter for Rs 22000. He pays Rs 4000 cash and agrees to pay the balance in annual instalments of Rs 1000 plus 10% interest on the unpaid amount. How much the scooter will cost him.
Write the common difference of an A.P. whose nth term is xn + y.
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.
Write the sum of first n odd natural numbers.
If \[\frac{3 + 5 + 7 + . . . + \text { upto n terms }}{5 + 8 + 11 + . . . . \text { upto 10 terms }}\] 7, then find the value of n.
Sum of all two digit numbers which when divided by 4 yield unity as remainder is
Mark the correct alternative in the following question:
\[\text { If in an A . P } . S_n = n^2 q \text { and } S_m = m^2 q, \text { where } S_r \text{ denotes the sum of r terms of the A . P . , then }S_q \text { equals }\]
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
If a, b, c, d are four distinct positive quantities in A.P., then show that bc > ad
The first term of an A.P.is a, and the sum of the first p terms is zero, show that the sum of its next q terms is `(-a(p + q)q)/(p - 1)`
A man accepts a position with an initial salary of Rs 5200 per month. It is understood that he will receive an automatic increase of Rs 320 in the very next month and each month thereafter. What is his total earnings during the first year?
If the sum of p terms of an A.P. is q and the sum of q terms is p, show that the sum of p + q terms is – (p + q). Also, find the sum of first p – q terms (p > q).
If the sum of n terms of an A.P. is given by Sn = 3n + 2n2, then the common difference of the A.P. is ______.
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 ______.
Let Sn denote the sum of the first n terms of an A.P. If S2n = 3Sn then S3n: Sn is equal to ______.
If n AM's are inserted between 1 and 31 and ratio of 7th and (n – 1)th A.M. is 5:9, then n equals ______.
If the first term of an A.P. is 3 and the sum of its first 25 terms is equal to the sum of its next 15 terms, then the common difference 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 ______.
If b2, a2, c2 are in A.P., then `1/(a + b), 1/(b + c), 1/(c + a)` will be in ______
