Advertisements
Advertisements
प्रश्न
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.
Advertisements
उत्तर
Since
\[\frac{b + c}{a}, \frac{c + a}{b}, \frac{a + b}{c}\] are in A.P., we have:
\[\frac{c + a}{b} - \frac{b + c}{a} = \frac{a + b}{c} - \frac{c + a}{b}\]
\[ \Rightarrow \frac{ac + a^2 - b^2 - bc}{ab} = \frac{ab + b^2 - c^2 - ac}{bc}\]
\[ \Rightarrow \frac{\left( a + b \right)\left( a - b \right) + c\left( a - b \right)}{ab} = \frac{\left( b + c \right)\left( b - c \right) + a\left( b - c \right)}{bc}\]
\[ \Rightarrow \frac{\left( a - b \right)\left( a + b + c \right)}{ab} = \frac{\left( b - c \right)\left( a + b + c \right)}{bc}\]
\[ \Rightarrow \frac{\left( a - b \right)}{ab} = \frac{\left( b - c \right)}{bc}\]
\[ \Rightarrow \frac{1}{b} - \frac{1}{a} = \frac{1}{c} - \frac{1}{b}\]
Hence,
\[\frac{1}{a}, \frac{1}{b}, \frac{1}{c}\] are in A.P.
APPEARS IN
संबंधित प्रश्न
Find the sum of odd integers from 1 to 2001.
If the sum of a certain number of terms of the A.P. 25, 22, 19, … is 116. Find the last term
The sums of n terms of two arithmetic progressions are in the ratio 5n + 4: 9n + 6. Find the ratio of their 18th terms
Insert five numbers between 8 and 26 such that the resulting sequence is an A.P.
A man starts repaying a loan as first installment of Rs. 100. If he increases the installment by Rs 5 every month, what amount he will pay in the 30th installment?
Find the sum of all two digit numbers which when divided by 4, yields 1 as remainder.
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.
Show that the following sequence is an A.P. Also find the common difference and write 3 more terms in case.
3, −1, −5, −9 ...
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.
Find the second term and nth term of an A.P. whose 6th term is 12 and the 8th term is 22.
How many numbers are there between 1 and 1000 which when divided by 7 leave remainder 4?
Three numbers are in A.P. If the sum of these numbers be 27 and the product 648, find the numbers.
Find the sum of the following arithmetic progression :
41, 36, 31, ... to 12 terms
Find the sum of all integers between 84 and 719, which are multiples of 5.
Find the sum of the series:
3 + 5 + 7 + 6 + 9 + 12 + 9 + 13 + 17 + ... to 3n terms.
Find the r th term of an A.P., the sum of whose first n terms is 3n2 + 2n.
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 odd integers from 1 to 2001.
A man accepts a position with an initial salary of ₹5200 per month. It is understood that he will receive an automatic increase of ₹320 in the very next month and each month thereafter.
(i) Find his salary for the tenth month.
(ii) What is his total earnings during the first year?
A man saved ₹66000 in 20 years. In each succeeding year after the first year he saved ₹200 more than what he saved in the previous year. How much did he save in the first year?
Write the sum of first n even natural numbers.
If m th term of an A.P. is n and nth term is m, then write its pth term.
If the sums of n terms of two AP.'s are in the ratio (3n + 2) : (2n + 3), then find the ratio of their 12th terms.
If Sn denotes the sum of first n terms of an A.P. < an > such that
If the sum of n terms of an A.P., is 3 n2 + 5 n then which of its terms is 164?
In the arithmetic progression whose common difference is non-zero, the sum of first 3 n terms is equal to the sum of next n terms. Then the ratio of the sum of the first 2 n terms to the next 2 nterms is
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 second, third and sixth terms of an A.P. are consecutive terms of a G.P., write the common ratio of the G.P.
Write the quadratic equation the arithmetic and geometric means of whose roots are Aand G respectively.
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
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 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?
Find the rth term of an A.P. sum of whose first n terms is 2n + 3n2
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 ______.
Let Sn denote the sum of the first n terms of an A.P. If S2n = 3Sn then S3n: Sn is equal to ______.
The sum of terms equidistant from the beginning and end in an A.P. is equal to ______.
