Advertisements
Advertisements
प्रश्न
If a, b, c are in G.P., prove that log a, log b, log c are in A.P.
उत्तर
a ,b and c are in G.P.
\[\therefore b^2 = ac\]
\[\text { Now, taking log on both the sides: } \]
\[ \Rightarrow \log \left( b \right)^2 = \log ac \]
\[ \Rightarrow 2\log b = \log a + \log c\]
\[\text { Thus, }\log a, \log b \text { and } \log\text { c are in A . P } .\]
APPEARS IN
संबंधित प्रश्न
For what values of x, the numbers `-2/7, x, -7/2` are in G.P?
Find the sum to 20 terms in the geometric progression 0.15, 0.015, 0.0015,…
If the pth , qth and rth terms of a G.P. are a, b and c, respectively. Prove that `a^(q - r) b^(r-p) c^(p-q) = 1`
If a, b, c, d are in G.P, prove that (an + bn), (bn + cn), (cn + dn) are in G.P.
Show that one of the following progression is a G.P. Also, find the common ratio in case:
\[a, \frac{3 a^2}{4}, \frac{9 a^3}{16}, . . .\]
Show that the sequence <an>, defined by an = \[\frac{2}{3^n}\], n ϵ N is a G.P.
Find :
the 12th term of the G.P.
\[\frac{1}{a^3 x^3}, ax, a^5 x^5 , . . .\]
If 5th, 8th and 11th terms of a G.P. are p. q and s respectively, prove that q2 = ps.
Find three numbers in G.P. whose sum is 65 and whose product is 3375.
Find the sum of the following geometric progression:
1, −1/2, 1/4, −1/8, ... to 9 terms;
Find the sum of the following geometric series:
\[\sqrt{2} + \frac{1}{\sqrt{2}} + \frac{1}{2\sqrt{2}} + . . .\text { to 8 terms };\]
Find the sum of the following geometric series:
\[\frac{2}{9} - \frac{1}{3} + \frac{1}{2} - \frac{3}{4} + . . . \text { to 5 terms };\]
Find the sum of the following geometric series:
\[\frac{a}{1 + i} + \frac{a}{(1 + i )^2} + \frac{a}{(1 + i )^3} + . . . + \frac{a}{(1 + i )^n} .\]
The sum of n terms of the G.P. 3, 6, 12, ... is 381. Find the value of n.
Find the sum of the following serie to infinity:
\[1 - \frac{1}{3} + \frac{1}{3^2} - \frac{1}{3^3} + \frac{1}{3^4} + . . . \infty\]
Prove that: (91/3 . 91/9 . 91/27 ... ∞) = 3.
Find the rational numbers having the following decimal expansion:
\[0 .\overline {231 }\]
The sum of three numbers in G.P. is 56. If we subtract 1, 7, 21 from these numbers in that order, we obtain an A.P. Find the numbers.
If a, b, c are in G.P., prove that:
\[a^2 b^2 c^2 \left( \frac{1}{a^3} + \frac{1}{b^3} + \frac{1}{c^3} \right) = a^3 + b^3 + c^3\]
If a, b, c are in G.P., prove that:
(a + 2b + 2c) (a − 2b + 2c) = a2 + 4c2.
If a, b, c, d are in G.P., prove that:
(a2 + b2 + c2), (ab + bc + cd), (b2 + c2 + d2) are in G.P.
The sum of an infinite G.P. is 4 and the sum of the cubes of its terms is 92. The common ratio of the original G.P. is
If A be one A.M. and p, q be two G.M.'s between two numbers, then 2 A is equal to
The number of bacteria in a culture doubles every hour. If there were 50 bacteria originally in the culture, how many bacteria will be there at the end of 5thhour?
Mosquitoes are growing at a rate of 10% a year. If there were 200 mosquitoes in the beginning. Write down the number of mosquitoes after n years.
For a sequence, if Sn = 2(3n –1), find the nth term, hence show that the sequence is a G.P.
If Sn, S2n, S3n are the sum of n, 2n, 3n terms of a G.P. respectively, then verify that Sn (S3n – S2n) = (S2n – Sn)2.
Find: `sum_("r" = 1)^10(3 xx 2^"r")`
Express the following recurring decimal as a rational number:
`0.bar(7)`
Find : `sum_("r" = 1)^oo 4(0.5)^"r"`
Find : `sum_("r" = 1)^oo (-1/3)^"r"`
Insert two numbers between 1 and −27 so that the resulting sequence is a G.P.
Select the correct answer from the given alternative.
The tenth term of the geometric sequence `1/4, (-1)/2, 1, -2,` ... is –
Select the correct answer from the given alternative.
Which term of the geometric progression 1, 2, 4, 8, ... is 2048
Answer the following:
Find the nth term of the sequence 0.6, 0.66, 0.666, 0.6666, ...
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 ______.
Let S be the sum, P be the product and R be the sum of the reciprocals of 3 terms of a G.P. Then P2 R3 : S3 is equal to ______.
If the pth and qth terms of a G.P. are q and p respectively, show that its (p + q)th term is `(q^p/p^q)^(1/(p - q))`
If `e^((cos^2x + cos^4x + cos^6x + ...∞)log_e2` satisfies the equation t2 – 9t + 8 = 0, then the value of `(2sinx)/(sinx + sqrt(3)cosx)(0 < x ,< π/2)` is ______.
