Advertisements
Advertisements
प्रश्न
Find the geometric means of the following pairs of number:
2 and 8
उत्तर
\[\text { Let the G . M . between 2 and 8 be G } . \]
\[\text { Then, 2, G and 8 are in G . P } . \]
\[ \therefore G^2 = 2 \times 8\]
\[ \Rightarrow G^2 = 16\]
\[ \Rightarrow G = \pm \sqrt{16}\]
\[ \Rightarrow G = \pm 4\]
APPEARS IN
संबंधित प्रश्न
Which term of the following sequence:
`1/3, 1/9, 1/27`, ...., is `1/19683`?
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.
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.
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.
Find :
the 12th term of the G.P.
\[\frac{1}{a^3 x^3}, ax, a^5 x^5 , . . .\]
The sum of first three terms of a G.P. is \[\frac{39}{10}\] and their product is 1. Find the common ratio and the terms.
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.
Find the sum of the following geometric series:
`3/5 + 4/5^2 + 3/5^3 + 4/5^4 + ....` to 2n terms;
How many terms of the G.P. 3, 3/2, 3/4, ... be taken together to make \[\frac{3069}{512}\] ?
The common ratio of a G.P. is 3 and the last term is 486. If the sum of these terms be 728, find the first term.
The ratio of the sum of first three terms is to that of first 6 terms of a G.P. is 125 : 152. Find the common ratio.
The fifth term of a G.P. is 81 whereas its second term is 24. Find the series and sum of its first eight terms.
If a and b are the roots of x2 − 3x + p = 0 and c, d are the roots x2 − 12x + q = 0, where a, b, c, d form a G.P. Prove that (q + p) : (q − p) = 17 : 15.
A G.P. consists of an even number of terms. If the sum of all the terms is 5 times the sum of the terms occupying the odd places. Find the common ratio of the G.P.
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
\[\frac{2S S_1}{S^2 + S_1}\text { and } \frac{S^2 - S_1}{S^2 + S_1}\]
Three numbers are in A.P. and their sum is 15. If 1, 3, 9 be added to them respectively, they form 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:
(a + b + c + d)2 = (a + b)2 + 2 (b + c)2 + (c + d)2
If a, b, c are in G.P., prove that the following is also in G.P.:
a3, b3, c3
If \[\frac{1}{a + b}, \frac{1}{2b}, \frac{1}{b + c}\] are three consecutive terms of an A.P., prove that a, b, c are the three consecutive terms of a 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 the sum of an infinite decreasing G.P. is 3 and the sum of the squares of its term is \[\frac{9}{2}\], then write its first term and common difference.
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
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
For the G.P. if r = `1/3`, a = 9 find t7
For the G.P. if a = `7/243`, r = 3 find t6.
Find three numbers in G.P. such that their sum is 21 and sum of their squares is 189.
Find four numbers in G.P. such that sum of the middle two numbers is `10/3` and their product is 1
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?
For a G.P. If t3 = 20 , t6 = 160 , find S7
For a G.P. If t4 = 16, t9 = 512, find S10
Find the sum to n terms of the sequence.
0.5, 0.05, 0.005, ...
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.
Determine whether the sum to infinity of the following G.P.s exist, if exists find them:
`1/2, 1/4, 1/8, 1/16,...`
Determine whether the sum to infinity of the following G.P.s exist, if exists find them:
`-3, 1, (-1)/3, 1/9, ...`
Express the following recurring decimal as a rational number:
`2.bar(4)`
Find : `sum_("r" = 1)^oo 4(0.5)^"r"`
Answer the following:
Find the sum of infinite terms of `1 + 4/5 + 7/25 + 10/125 + 13/6225 + ...`
