Advertisements
Advertisements
Question
If a, b, c are in G.P., prove that \[\frac{1}{\log_a m}, \frac{1}{\log_b m}, \frac{1}{\log_c m}\] are in A.P.
Solution
a, b, c are in G.P.
\[\therefore b^2 = ac \]
\[\text { Now taking } lo g_m \text { on both the sides: } \]
\[ \Rightarrow lo g_m \left( b \right)^2 = lo g_m \left( ac \right)\]
\[ \Rightarrow 2lo g_m \left( b \right) = lo g_m a + lo g_m \left( c \right)\]
\[ \Rightarrow \frac{2}{\log_b \left( m \right)} = \frac{1}{\log_a \left( m \right)} + \frac{1}{\log_c \left( m \right)}\]
\[\text { Thus }, \frac{1}{\log_a \left( m \right)}, \frac{1}{\log_b \left( m \right)} \text { and } \frac{1}{\log_c \left( m \right)} \text { are in A . P } . \]
APPEARS IN
RELATED QUESTIONS
The 5th, 8th and 11th terms of a G.P. are p, q and s, respectively. Show that q2 = ps.
Which term of the following sequence:
`1/3, 1/9, 1/27`, ...., is `1/19683`?
For what values of x, the numbers `-2/7, x, -7/2` are in G.P?
Find the sum of the products of the corresponding terms of the sequences `2, 4, 8, 16, 32 and 128, 32, 8, 2, 1/2`
If f is a function satisfying f (x +y) = f(x) f(y) for all x, y ∈ N such that f(1) = 3 and `sum_(x = 1)^n` f(x) = 120, find the value of n.
The first term of a G.P. is 1. The sum of the third term and fifth term is 90. Find the common ratio of G.P.
if `(a+ bx)/(a - bx) = (b +cx)/(b - cx) = (c + dx)/(c- dx) (x != 0)` then show that a, b, c and d are in G.P.
If a, b, c, d are in G.P, prove that (an + bn), (bn + cn), (cn + dn) are in G.P.
Show that the sequence <an>, defined by an = \[\frac{2}{3^n}\], n ϵ N is a G.P.
Find the sum of the following geometric series:
`3/5 + 4/5^2 + 3/5^3 + 4/5^4 + ....` to 2n terms;
Evaluate the following:
\[\sum^{10}_{n = 2} 4^n\]
Find the sum of the following series:
9 + 99 + 999 + ... to n terms;
How many terms of the G.P. 3, 3/2, 3/4, ... be taken together to make \[\frac{3069}{512}\] ?
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: (21/4 . 41/8 . 81/16. 161/32 ... ∞) = 2.
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 the following is also in G.P.:
a3, b3, c3
If pth, qth, rth and sth terms of an A.P. be in G.P., then prove that p − q, q − r, r − s are in G.P.
If a, b, c are in A.P., b,c,d are in G.P. and \[\frac{1}{c}, \frac{1}{d}, \frac{1}{e}\] are in A.P., prove that a, c,e are in G.P.
If logxa, ax/2 and logb x are in G.P., then write the value of x.
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 x is positive, the sum to infinity of the series \[\frac{1}{1 + x} - \frac{1 - x}{(1 + x )^2} + \frac{(1 - x )^2}{(1 + x )^3} - \frac{(1 - x )^3}{(1 + x )^4} + . . . . . . is\]
Let x be the A.M. and y, z be two G.M.s between two positive numbers. Then, \[\frac{y^3 + z^3}{xyz}\] is equal to
Check whether the following sequence is G.P. If so, write tn.
3, 4, 5, 6, …
The numbers 3, x, and x + 6 form are in G.P. Find 20th term.
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.
Express the following recurring decimal as a rational number:
`2.3bar(5)`
The midpoints of the sides of a square of side 1 are joined to form a new square. This procedure is repeated indefinitely. Find the sum of the areas of all the squares
Select the correct answer from the given alternative.
Which term of the geometric progression 1, 2, 4, 8, ... is 2048
Select the correct answer from the given alternative.
If common ratio of the G.P is 5, 5th term is 1875, the first term is -
Answer the following:
For a sequence , if tn = `(5^("n" - 2))/(7^("n" - 3))`, verify whether the sequence is a G.P. If it is a G.P., find its first term and the common ratio.
If a, b, c, d are four distinct positive quantities in G.P., then show that a + d > b + c
The sum or difference of two G.P.s, is again a G.P.
The sum of the first three terms of a G.P. is S and their product is 27. Then all such S lie in ______.
