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.
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.
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.
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?
Show that the sum of (m + n)th and (m – n)th terms of an A.P. is equal to twice the mth term.
If the sum of three numbers in A.P., is 24 and their product is 440, find the numbers.
Find the sum of all two digit numbers which when divided by 4, yields 1 as remainder.
Let < an > be a sequence defined by a1 = 3 and, an = 3an − 1 + 2, for all n > 1
Find the first four terms of the sequence.
Show that the following sequence is an A.P. Also find the common difference and write 3 more terms in case.
\[\sqrt{2}, 3\sqrt{2}, 5\sqrt{2}, 7\sqrt{2}, . . .\]
The nth term of a sequence is given by an = 2n + 7. Show that it is an A.P. Also, find its 7th term.
If the sequence < an > is an A.P., show that am +n +am − n = 2am.
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.
The sum of three terms of an A.P. is 21 and the product of the first and the third terms exceeds the second term by 6, find three terms.
Find the four numbers in A.P., whose sum is 50 and in which the greatest number is 4 times the least.
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 first n natural numbers.
Find the sum of all odd numbers between 100 and 200.
Find the sum of the series:
3 + 5 + 7 + 6 + 9 + 12 + 9 + 13 + 17 + ... to 3n terms.
Find the sum of all those integers between 100 and 800 each of which on division by 16 leaves the remainder 7.
Find the r th term of an A.P., the sum of whose first n terms is 3n2 + 2n.
The sum of first 7 terms of an A.P. is 10 and that of next 7 terms is 17. Find the progression.
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.
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 \[a\left( \frac{1}{b} + \frac{1}{c} \right), b\left( \frac{1}{c} + \frac{1}{a} \right), c\left( \frac{1}{a} + \frac{1}{b} \right)\] are in A.P., prove that a, b, c are in A.P.
A man arranges to pay off a debt of Rs 3600 by 40 annual instalments which form an arithmetic series. When 30 of the instalments are paid, he dies leaving one-third of the debt unpaid, find the value of the first instalment.
A farmer buys a used tractor for Rs 12000. He pays Rs 6000 cash and agrees to pay the balance in annual instalments of Rs 500 plus 12% interest on the unpaid amount. How much the tractor cost him?
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?
Write the common difference of an A.P. the sum of whose first n terms is
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
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 the sum of first n even natural numbers is equal to k times the sum of first n odd natural numbers, then k =
If the first, second and last term of an A.P are a, b and 2a respectively, then its sum is
Mark the correct alternative in the following question:
If in an A.P., the pth term is q and (p + q)th term is zero, then the qth term 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 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. Find his salary for the tenth month
If 9 times the 9th term of an A.P. is equal to 13 times the 13th term, then the 22nd term of the A.P. is ______.
