Advertisements
Advertisements
Question
The two geometric means between the numbers 1 and 64 are
Options
(a) 1 and 64
(b) 4 and 16
(c) 2 and 16
(d) 8 and 16
(e) 3 and 16
Advertisements
Solution
(b) 4 and 16
\[\text{ Let the two G . M . s between 1 and 64 be G_1 and G_2 } . \]
\[\text{ Thus, 1, G_1 , G_2 and 64 are in G . P } . \]
\[ 64 = 1 \times r^3 \]
\[ \Rightarrow r = \sqrt[3]{64}\]
\[ \Rightarrow r = 4\]
\[ \Rightarrow G_1 = ar = 1 \times 4 = 4\]
\[\text{ And }, G_2 = a r^2 = 1 \times 4^2 = 16\]
\[\text{ Thus, 4 and 16 are the required G . M . s } .\]
APPEARS IN
RELATED QUESTIONS
Find a G.P. for which sum of the first two terms is –4 and the fifth term is 4 times the third term.
Show that the products of the corresponding terms of the sequences a, ar, ar2, …arn – 1 and A, AR, AR2, … `AR^(n-1)` form a G.P, and find the common ratio
Find the value of n so that `(a^(n+1) + b^(n+1))/(a^n + b^n)` may be the geometric mean between a and b.
The sum of two numbers is 6 times their geometric mean, show that numbers are in the ratio `(3 + 2sqrt2) ":" (3 - 2sqrt2)`.
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.
\[- \frac{3}{4}, \frac{1}{2}, - \frac{1}{3}, \frac{2}{9}, . . .\]
Find :
the 12th term of the G.P.
\[\frac{1}{a^3 x^3}, ax, a^5 x^5 , . . .\]
Which term of the G.P.: `sqrt3, 3, 3sqrt3`, ... is 729?
In a GP the 3rd term is 24 and the 6th term is 192. Find the 10th term.
The sum of three numbers in G.P. is 14. If the first two terms are each increased by 1 and the third term decreased by 1, the resulting numbers are in A.P. Find the numbers.
How many terms of the series 2 + 6 + 18 + ... must be taken to make the sum equal to 728?
Find the sum of the following serie to infinity:
\[1 - \frac{1}{3} + \frac{1}{3^2} - \frac{1}{3^3} + \frac{1}{3^4} + . . . \infty\]
Find the sum of the following series to infinity:
10 − 9 + 8.1 − 7.29 + ... ∞
One side of an equilateral triangle is 18 cm. The mid-points of its sides are joined to form another triangle whose mid-points, in turn, are joined to form still another triangle. The process is continued indefinitely. Find the sum of the (i) perimeters of all the triangles. (ii) areas of all triangles.
The sum of three numbers which are consecutive terms of an A.P. is 21. If the second number is reduced by 1 and the third is increased by 1, we obtain three consecutive terms of a G.P. Find the numbers.
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, d are in G.P., prove that:
(a + b + c + d)2 = (a + b)2 + 2 (b + c)2 + (c + d)2
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 a, b, c are in A.P. and a, b, d are in G.P., show that a, (a − b), (d − c) are in G.P.
Insert 6 geometric means between 27 and \[\frac{1}{81}\] .
Find the geometric means of the following pairs of number:
a3b and ab3
The sum of two numbers is 6 times their geometric means, show that the numbers are in the ratio `(3+2sqrt2):(3-2sqrt2)`.
In a G.P. if the (m + n)th term is p and (m − n)th term is q, then its mth term is
Which term of the G.P. 5, 25, 125, 625, … is 510?
The numbers 3, x, and x + 6 form are in G.P. Find 20th term.
The numbers x − 6, 2x and x2 are in G.P. Find nth term
For a G.P. If t4 = 16, t9 = 512, find S10
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
Determine whether the sum to infinity of the following G.P.s exist, if exists find them:
`2, 4/3, 8/9, 16/27, ...`
If the first term of the G.P. is 16 and its sum to infinity is `96/17` find the common ratio.
Find : `sum_("r" = 1)^oo (-1/3)^"r"`
Select the correct answer from the given alternative.
If for a G.P. `"t"_6/"t"_3 = 1458/54` then r = ?
Answer the following:
Find the sum of the first 5 terms of the G.P. whose first term is 1 and common ratio is `2/3`
If a, b, c, d are four distinct positive quantities in G.P., then show that a + d > b + c
For a, b, c to be in G.P. the value of `(a - b)/(b - c)` is equal to ______.
If in a geometric progression {an}, a1 = 3, an = 96 and Sn = 189, then the value of n is ______.
If 0 < x, y, a, b < 1, then the sum of the infinite terms of the series `sqrt(x)(sqrt(a) + sqrt(x)) + sqrt(x)(sqrt(ab) + sqrt(xy)) + sqrt(x)(bsqrt(a) + ysqrt(x)) + ...` 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 ______.
