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
\[\frac{2n}{n + 1}\]
\[\frac{n}{n + 1}\]
\[\frac{n + 1}{2n}\]
\[\frac{n + 1}{n}\]
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
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 the sum of first p terms of an A.P. is equal to the sum of the first q terms, then find the sum of the first (p + q) terms.
Between 1 and 31, m numbers have been inserted in such a way that the resulting sequence is an A.P. and the ratio of 7th and (m – 1)th numbers is 5:9. Find the value of m.
If the sum of three numbers in A.P., is 24 and their product is 440, find the numbers.
Let the sum of n, 2n, 3n terms of an A.P. be S1, S2 and S3, respectively, show that S3 = 3 (S2– S1)
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 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
A person writes a letter to four of his friends. He asks each one of them to copy the letter and mail to four different persons with instruction that they move the chain similarly. Assuming that the chain is not broken and that it costs 50 paise to mail one letter. Find the amount spent on the postage when 8th set of letter is mailed.
Let < an > be a sequence. Write the first five term in the following:
a1 = a2 = 2, an = an − 1 − 1, n > 2
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 = 2n2 + n + 1. Show that it is not an A.P.
Find:
18th term of the A.P.
\[\sqrt{2}, 3\sqrt{2}, 5\sqrt{2},\]
Which term of the sequence 12 + 8i, 11 + 6i, 10 + 4i, ... is purely imaginary?
If 9th term of an A.P. is zero, prove that its 29th term is double the 19th term.
The 10th and 18th terms of an A.P. are 41 and 73 respectively. Find 26th 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.
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 :
3, 9/2, 6, 15/2, ... to 25 terms
Find the sum of the following arithmetic progression :
41, 36, 31, ... to 12 terms
Find the sum of first n natural numbers.
Find the sum of all natural numbers between 1 and 100, which are divisible by 2 or 5.
Find the sum of all odd numbers between 100 and 200.
Show that the sum of all odd integers between 1 and 1000 which are divisible by 3 is 83667.
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 \[a\left( \frac{1}{b} + \frac{1}{c} \right), b\left( \frac{1}{c} + \frac{1}{a} \right), c\left( \frac{1}{a} + \frac{1}{b} \right)\] are in A.P., prove that a, b, c are in A.P.
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.
If \[\frac{3 + 5 + 7 + . . . + \text { upto n terms }}{5 + 8 + 11 + . . . . \text { upto 10 terms }}\] 7, then find the value of n.
If the sum of n terms of an A.P., is 3 n2 + 5 n then which of its terms is 164?
If the sum of n terms of an A.P. is 2 n2 + 5 n, then its nth term is
If a1, a2, a3, .... an are in A.P. with common difference d, then the sum of the series sin d [cosec a1cosec a2 + cosec a1 cosec a3 + .... + cosec an − 1 cosec an] is
If four numbers in A.P. are such that their sum is 50 and the greatest number is 4 times the least, then the numbers are
If in an A.P., Sn = n2p and Sm = m2p, where Sr denotes the sum of r terms of the A.P., then Sp is equal to
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 pth term of an A.P. is a and qth term is b. Prove that the sum of its (p + q) terms is `(p + q)/2[a + b + (a - b)/(p - q)]`.
In an A.P. the pth term is q and the (p + q)th term is 0. Then the qth term is ______.
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 ______.
