Advertisements
Advertisements
प्रश्न
If \[\frac{1}{a}, \frac{1}{b}, \frac{1}{c}\] are in A.P., prove that:
\[\frac{b + c}{a}, \frac{c + a}{b}, \frac{a + b}{c}\] are in A.P.
Advertisements
उत्तर
\[\text { Given }: \frac{1}{a}, \frac{1}{b}, \frac{1}{c} \text { are in A . P } . \]
\[ \therefore \frac{2}{b} = \frac{1}{a} + \frac{1}{c}\]
\[ \Rightarrow 2ac = ab + bc . . . . (1)\]
\[\text { To prove }: \frac{b + c}{a}, \frac{c + a}{b}, \frac{a + b}{c} \text { are in A . P } . \]
\[ 2\left( \frac{a + c}{b} \right) = \frac{b + c}{a} + \frac{a + b}{c}\]
\[ \Rightarrow 2ac(a + c) = bc(b + c) + ab(a + b)\]
\[\text { LHS: }2ac(a + c)\]
\[ = (ab + bc)(a + c) (\text { From }(1))\]
\[ = a^2 b + 2abc + b c^2 \]
\[\text { RHS: } bc(b + c) + ab(a + b)\]
\[ = b^2 c + b c^2 + a^2 b + a b^2 \]
\[ = b^2 c + a b^2 + b c^2 + a^2 b\]
\[ = b(bc + ab) + b c^2 + a^2 b\]
\[ = 2abc + b c^2 + a^2 b \]
\[ = a^2 b + 2abc + b c^2 (\text { From }(1))\]
\[ \therefore \text { LHS = RHS }\]
\[\text { Hence, proved } . \]
APPEARS IN
संबंधित प्रश्न
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.
Sum of the first p, q and r terms of an A.P. are a, b and c, respectively.
Prove that `a/p (q - r) + b/q (r- p) + c/r (p - q) = 0`
if `(a^n + b^n)/(a^(n-1) + b^(n-1))` is the A.M. between a and b, then find the value of n.
If the sum of three numbers in A.P., is 24 and their product is 440, find the numbers.
Find the sum of integers from 1 to 100 that are divisible by 2 or 5.
if `a(1/b + 1/c), b(1/c+1/a), c(1/a+1/b)` are in A.P., prove that a, b, c are in A.P.
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.
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.
Find:
10th term of the A.P. 1, 4, 7, 10, ...
Find:
18th term of the A.P.
\[\sqrt{2}, 3\sqrt{2}, 5\sqrt{2},\]
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:
1, 4, 7, 10, ..., 88
An A.P. consists of 60 terms. If the first and the last terms be 7 and 125 respectively, find 32nd term.
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 :
41, 36, 31, ... to 12 terms
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?
Find the sum of n terms of the A.P. whose kth terms is 5k + 1.
Find the sum of odd integers from 1 to 2001.
If S1 be the sum of (2n + 1) terms of an A.P. and S2 be the sum of its odd terms, then prove that: S1 : S2 = (2n + 1) : (n + 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.
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 is employed to count Rs 10710. He counts at the rate of Rs 180 per minute for half an hour. After this he counts at the rate of Rs 3 less every minute than the preceding minute. Find the time taken by him to count the entire amount.
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 p terms of an A.P. is q and the sum of q terms is p, then the sum of p + q terms will be
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
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 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 second, third and sixth terms of an A.P. are consecutive terms of a G.P., write the common ratio of the G.P.
If a, b, c are in G.P. and a1/x = b1/y = c1/z, then xyz are in
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 product of three numbers in A.P. is 224, and the largest number is 7 times the smallest. Find the numbers
If the sum of m terms of an A.P. is equal to the sum of either the next n terms or the next p terms, then prove that `(m + n) (1/m - 1/p) = (m + p) (1/m - 1/n)`
If a1, a2, ..., an are in A.P. with common difference d (where d ≠ 0); then the sum of the series sin d (cosec a1 cosec a2 + cosec a2 cosec a3 + ...+ cosec an–1 cosec an) is equal to cot a1 – cot an
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. Find his salary for the tenth month
The sum of terms equidistant from the beginning and end in an A.P. is equal to ______.
If the sum of n terms of a sequence is quadratic expression then it always represents an A.P
