Advertisements
Advertisements
प्रश्न
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
Advertisements
उत्तर
Let a be the first term and d the common difference of the A.P
Also let S1 be the sum of odd terms of A.P. having (2n + 1) terms.
Then S1 = a1 + a3 + a5 + ... + a2n+1
S1 = `(n + 1)/2 (a_1 + a_(2n + 1))`
S1 = `(n + 1)/2 [a + a + (2n + 1 - 1)d]`
= (n + 1) (a + nd)
Similarly, if S2 denotes the sum of even terms, then
S2 = `n/2 [2a + 2nd]` = n(a + nd)
Hence, `"S"_1/"S"_2 = ((n + 1)(a + nd))/(n(a + nd))`
= `(n + 1)/n`
APPEARS IN
संबंधित प्रश्न
If the sum of a certain number of terms of the A.P. 25, 22, 19, … is 116. Find the last term
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 difference between any two consecutive interior angles of a polygon is 5°. If the smallest angle is 120°, find the number of the sides of the polygon.
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.
Shamshad Ali buys a scooter for Rs 22000. He pays Rs 4000 cash and agrees to pay the balance in annual installment of Rs 1000 plus 10% interest on the unpaid amount. How much will the scooter cost him?
A sequence is defined by an = n3 − 6n2 + 11n − 6, n ϵ N. Show that the first three terms of the sequence are zero and all other terms are positive.
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, ...
Which term of the A.P. 3, 8, 13, ... is 248?
Is 302 a term of the A.P. 3, 8, 13, ...?
The 10th and 18th terms of an A.P. are 41 and 73 respectively. Find 26th term.
Find the second term and nth term of an A.P. whose 6th term is 12 and the 8th term is 22.
The angles of a quadrilateral are in A.P. whose common difference is 10°. Find the angles.
Find the sum of all integers between 84 and 719, which are multiples of 5.
Solve:
25 + 22 + 19 + 16 + ... + x = 115
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 the sum of a certain number of terms of the AP 25, 22, 19, ... is 116. Find the last term.
The sums of first n terms of two A.P.'s are in the ratio (7n + 2) : (n + 4). Find the ratio of their 5th 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 a, b, c is in A.P., then show that:
bc − a2, ca − b2, ab − c2 are in 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 starts repaying a loan as first instalment of Rs 100 = 00. If he increases the instalments by Rs 5 every month, what amount he will pay in the 30th instalment?
If log 2, log (2x − 1) and log (2x + 3) are in A.P., write the value of x.
Write the value of n for which n th terms of the A.P.s 3, 10, 17, ... and 63, 65, 67, .... are equal.
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
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
If a1, a2, a3, .... an are in A.P. with common difference d, then the sum of the series sin d [sec a1 sec a2 + sec a2 sec a3 + .... + sec an − 1 sec 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, 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 \[\frac{S_1}{S_2}\] =
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 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 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
If a, b, c, d are four distinct positive quantities in A.P., then show that bc > ad
A man saved Rs 66000 in 20 years. In each succeeding year after the first year he saved Rs 200 more than what he saved in the previous year. How much did he save in 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).
