Advertisements
Advertisements
प्रश्न
If xa = xb/2 zb/2 = zc, then prove that \[\frac{1}{a}, \frac{1}{b}, \frac{1}{c}\] are in A.P.
उत्तर
\[\text { Here }, x^a = \left( xz \right)^\frac{b}{2} = z^c \]
\[\text { Now, taking log on both the sides: } \]
\[\log \left( x \right)^a = \log \left( xz \right)^\frac{b}{2} = \log \left( z \right)^c \]
\[ \Rightarrow \text { alog} x = \frac{b}{2} \log\left( xz \right) = c \log z\]
\[ \Rightarrow \text {alog } x = \frac{b}{2}\log x + \frac{b}{2}\log z = c \log z\]
\[ \Rightarrow \text { alog } x = \frac{b}{2}\log x + \frac{b}{2}\log z \text { and }\frac{b}{2}\log x + \frac{b}{2}\log z = c \log z\]
\[ \Rightarrow \left( a - \frac{b}{2} \right)\log x = \frac{b}{2} \log z \text { and }\frac{b}{2}\log x = \left( c - \frac{b}{2} \right)\log z\]
\[ \Rightarrow \frac{\log x}{\log z} = \frac{\frac{b}{2}}{\left( a - \frac{b}{2} \right)} \text { and } \frac{\log x}{\log z} = \frac{\left( c - \frac{b}{2} \right)}{\frac{b}{2}} \]
\[ \Rightarrow \frac{\frac{b}{2}}{\left( a - \frac{b}{2} \right)} = \frac{\left( c - \frac{b}{2} \right)}{\frac{b}{2}}\]
\[ \Rightarrow \frac{b^2}{4} = ac - \frac{ab}{2} - \frac{bc}{2} + \frac{b^2}{4}\]
\[ \Rightarrow 2ac = ab + bc\]
\[ \Rightarrow \frac{2}{b} = \frac{1}{a} + \frac{1}{c}\]
\[\text { Thus, } \frac{1}{a}, \frac{1}{b} \text { and } \frac{1}{c} \text { are in A . P } .\]
APPEARS IN
संबंधित प्रश्न
Find the 20th and nthterms of the G.P. `5/2, 5/4 , 5/8,...`
The 5th, 8th and 11th terms of a G.P. are p, q and s, respectively. Show that q2 = ps.
The 4th term of a G.P. is square of its second term, and the first term is –3. Determine its 7thterm.
For what values of x, the numbers `-2/7, x, -7/2` are in G.P?
Find the sum to indicated number of terms of the geometric progressions `sqrt7, sqrt21,3sqrt7`...n terms.
The sum of some terms of G.P. is 315 whose first term and the common ratio are 5 and 2, respectively. Find the last term and the number of terms.
A G.P. consists of an even number of terms. If the sum of all the terms is 5 times the sum of terms occupying odd places, then find its common ratio.
Find :
the 10th term of the G.P.
\[\sqrt{2}, \frac{1}{\sqrt{2}}, \frac{1}{2\sqrt{2}}, . . .\]
Which term of the G.P. :
\[2, 2\sqrt{2}, 4, . . .\text { is }128 ?\]
Which term of the progression 18, −12, 8, ... is \[\frac{512}{729}\] ?
Find the 4th term from the end of the G.P.
\[\frac{1}{2}, \frac{1}{6}, \frac{1}{18}, \frac{1}{54}, . . . , \frac{1}{4374}\]
In a GP the 3rd term is 24 and the 6th term is 192. Find the 10th term.
A person has 2 parents, 4 grandparents, 8 great grandparents, and so on. Find the number of his ancestors during the ten generations preceding his own.
Find the sum of the following series to infinity:
10 − 9 + 8.1 − 7.29 + ... ∞
Find an infinite G.P. whose first term is 1 and each term is the sum of all the terms which follow it.
If S denotes the sum of an infinite G.P. S1 denotes the sum of the squares of its terms, then prove that the first term and common ratio are respectively
\[\frac{2S S_1}{S^2 + S_1}\text { and } \frac{S^2 - S_1}{S^2 + S_1}\]
If a, b, c, d are in G.P., prove that:
(a + b + c + d)2 = (a + b)2 + 2 (b + c)2 + (c + d)2
If pth, qth and rth terms of a G.P. re x, y, z respectively, then write the value of xq − r yr − pzp − q.
If S be the sum, P the product and R be the sum of the reciprocals of n terms of a GP, then P2 is equal to
If second term of a G.P. is 2 and the sum of its infinite terms is 8, then its first term is
In a G.P. of even number of terms, the sum of all terms is five times the sum of the odd terms. The common ratio of the G.P. is
The numbers 3, x, and x + 6 form are in G.P. Find x
The numbers x − 6, 2x and x2 are in G.P. Find 1st term
The numbers x − 6, 2x and x2 are in G.P. Find nth term
For the following G.P.s, find Sn.
`sqrt(5)`, −5, `5sqrt(5)`, −25, ...
If S, P, R are the sum, product, and sum of the reciprocals of n terms of a G.P. respectively, then verify that `["S"/"R"]^"n"` = P2
Find: `sum_("r" = 1)^10 5 xx 3^"r"`
The value of a house appreciates 5% per year. How much is the house worth after 6 years if its current worth is ₹ 15 Lac. [Given: (1.05)5 = 1.28, (1.05)6 = 1.34]
Select the correct answer from the given alternative.
Sum to infinity of a G.P. 5, `-5/2, 5/4, -5/8, 5/16,...` is –
Select the correct answer from the given alternative.
Which of the following is not true, where A, G, H are the AM, GM, HM of a and b respectively. (a, b > 0)
Answer the following:
For a sequence Sn = 4(7n – 1) verify that the sequence is a G.P.
In a G.P. of positive terms, if any term is equal to the sum of the next two terms. Then the common ratio of the G.P. is ______.
If pth, qth, and rth terms of an A.P. and G.P. are both a, b and c respectively, show that ab–c . bc – a . ca – b = 1
The sum or difference of two G.P.s, is again a G.P.
If the sum of an infinite GP a, ar, ar2, ar3, ...... . is 15 and the sum of the squares of its each term is 150, then the sum of ar2, ar4, ar6, .... is ______.
If in a geometric progression {an}, a1 = 3, an = 96 and Sn = 189, then the value of n is ______.
Let A1, A2, A3, .... be an increasing geometric progression of positive real numbers. If A1A3A5A7 = `1/1296` and A2 + A4 = `7/36`, then the value of A6 + A8 + A10 is equal to ______.
