Advertisements
Advertisements
प्रश्न
The two geometric means between the numbers 1 and 64 are
पर्याय
(a) 1 and 64
(b) 4 and 16
(c) 2 and 16
(d) 8 and 16
(e) 3 and 16
Advertisements
उत्तर
(b) 4 and 16
\[\text{ Let the two G . M . s between 1 and 64 be G_1 and G_2 } . \]
\[\text{ Thus, 1, G_1 , G_2 and 64 are in G . P } . \]
\[ 64 = 1 \times r^3 \]
\[ \Rightarrow r = \sqrt[3]{64}\]
\[ \Rightarrow r = 4\]
\[ \Rightarrow G_1 = ar = 1 \times 4 = 4\]
\[\text{ And }, G_2 = a r^2 = 1 \times 4^2 = 16\]
\[\text{ Thus, 4 and 16 are the required G . M . s } .\]
संबंधित प्रश्न
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:
`2, 2sqrt2, 4,.... is 128`
Which term of the following sequence:
`1/3, 1/9, 1/27`, ...., is `1/19683`?
Find four numbers forming a geometric progression in which third term is greater than the first term by 9, and the second term is greater than the 4th by 18.
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.
Show that one of the following progression is a G.P. Also, find the common ratio in case:
−2/3, −6, −54, ...
Show that the sequence <an>, defined by an = \[\frac{2}{3^n}\], n ϵ N is a G.P.
Find :
the 10th term of the G.P.
\[\sqrt{2}, \frac{1}{\sqrt{2}}, \frac{1}{2\sqrt{2}}, . . .\]
Which term of the progression 18, −12, 8, ... is \[\frac{512}{729}\] ?
If 5th, 8th and 11th terms of a G.P. are p. q and s respectively, prove that q2 = ps.
Find the sum of the following geometric series:
0.15 + 0.015 + 0.0015 + ... to 8 terms;
Find the sum of the following series:
0.5 + 0.55 + 0.555 + ... to n terms.
How many terms of the series 2 + 6 + 18 + ... must be taken to make the sum equal to 728?
The ratio of the sum of the first three terms to that of the first 6 terms of a G.P. is 125 : 152. Find the common ratio.
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.
If a, b, c are in G.P., prove that:
\[\frac{(a + b + c )^2}{a^2 + b^2 + c^2} = \frac{a + b + c}{a - b + c}\]
If a, b, c, d are in G.P., prove that:
(b + c) (b + d) = (c + a) (c + d)
If a, b, c, d are in G.P., prove that:
(a2 + b2), (b2 + c2), (c2 + d2) are in G.P.
If a, b, c, d are in G.P., prove that:
(a2 − b2), (b2 − c2), (c2 − d2) are in G.P.
If a, b, c, d are in G.P., prove that:
\[\frac{1}{a^2 + b^2}, \frac{1}{b^2 - c^2}, \frac{1}{c^2 + d^2} \text { are in G . P } .\]
If (a − b), (b − c), (c − a) are in G.P., then prove that (a + b + c)2 = 3 (ab + bc + ca)
If a, b, c are in G.P., then prove that:
If pth, qth and rth terms of an A.P. and G.P. are both a, b and c respectively, show that \[a^{b - c} b^{c - a} c^{a - b} = 1\]
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 p, q be two A.M.'s and G be one G.M. between two numbers, then G2 =
Check whether the following sequence is G.P. If so, write tn.
1, –5, 25, –125 …
Check whether the following sequence is G.P. If so, write tn.
`sqrt(5), 1/sqrt(5), 1/(5sqrt(5)), 1/(25sqrt(5))`, ...
Check whether the following sequence is G.P. If so, write tn.
7, 14, 21, 28, …
For what values of x, the terms `4/3`, x, `4/27` are in G.P.?
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 5th hour?
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.
The numbers x − 6, 2x and x2 are in G.P. Find 1st term
Find the sum to n terms of the sequence.
0.5, 0.05, 0.005, ...
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 k so that k – 1, k, k + 2 are consecutive terms of a G.P.
Answer the following:
If for a G.P. first term is (27)2 and seventh term is (8)2, find S8
If x, 2y, 3z are in A.P., where the distinct numbers x, y, z are in G.P. then the common ratio of the G.P. is ______.
Let `{a_n}_(n = 0)^∞` be a sequence such that a0 = a1 = 0 and an+2 = 2an+1 – an + 1 for all n ≥ 0. Then, `sum_(n = 2)^∞ a^n/7^n` is equal to ______.
If the expansion in powers of x of the function `1/((1 - ax)(1 - bx))` is a0 + a1x + a2x2 + a3x3 ....... then an is ______.
