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
How many terms of the A.P. -6 , `-11/2` , -5... are needed to give the sum –25?
Find the sum to n terms of the A.P., whose kth term is 5k + 1.
If the sum of n terms of an A.P. is (pn + qn2), where p and q are constants, find the common difference.
Insert five numbers between 8 and 26 such that the resulting sequence is an A.P.
If the sum of three numbers in A.P., is 24 and their product is 440, find the numbers.
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
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.
A manufacturer reckons that the value of a machine, which costs him Rs 15625, will depreciate each year by 20%. Find the estimated value at the end of 5 years.
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.
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.
Which term of the A.P. 3, 8, 13, ... is 248?
Is 302 a term of the A.P. 3, 8, 13, ...?
The sum of 4th and 8th terms of an A.P. is 24 and the sum of the 6th and 10th terms is 34. Find the first term and the common difference of the A.P.
\[\text { If } \theta_1 , \theta_2 , \theta_3 , . . . , \theta_n \text { are in AP, whose common difference is d, then show that }\]
\[\sec \theta_1 \sec \theta_2 + \sec \theta_2 \sec \theta_3 + . . . + \sec \theta_{n - 1} \sec \theta_n = \frac{\tan \theta_n - \tan \theta_1}{\sin d} \left[ NCERT \hspace{0.167em} EXEMPLAR \right]\]
Find the sum of the following arithmetic progression :
1, 3, 5, 7, ... to 12 terms
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 all integers between 50 and 500 which are divisible by 7.
Solve:
1 + 4 + 7 + 10 + ... + x = 590.
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.
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 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 odd natural numbers.
If 7th and 13th terms of an A.P. be 34 and 64 respectively, then its 18th term 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
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, 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 \[\frac{S_1}{S_2}\] =
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.
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 ______.
If in an A.P., Sn = qn2 and Sm = qm2, where Sr denotes the sum of r terms of the A.P., then Sq equals ______.
Any term of an A.P. (except first) is equal to half the sum of terms which are equidistant from it.
Let 3, 6, 9, 12 ....... upto 78 terms and 5, 9, 13, 17 ...... upto 59 be two series. Then, the sum of the terms common to both the series is equal to ______.
If a1, a2, a3, .......... are an A.P. such that a1 + a5 + a10 + a15 + a20 + a24 = 225, then a1 + a2 + a3 + ...... + a23 + a24 is equal to ______.
The internal angles of a convex polygon are in A.P. The smallest angle is 120° and the common difference is 5°. The number to sides of the polygon is ______.
