Advertisements
Advertisements
प्रश्न
If the first term of a G.P. a1, a2, a3, ... is unity such that 4 a2 + 5 a3 is least, then the common ratio of G.P. is
विकल्प
−2/5
−3/5
2/5
none of these
Advertisements
उत्तर
− \[\frac{2}{5}\] If the first term is 1, then, the G.P. will be\[1, r, r^2 , r^3 , . . .\]
\[\text{ Now }, 5 r^2 + 4r = 5\left( r^2 + \frac{4}{5}r \right)\]
\[ = 5\left( r^2 + \frac{4}{5}r + \frac{4}{25} - \frac{4}{25} \right)\]
\[ = 5 \left( r + \frac{2}{5} \right)^2 - \frac{4}{5}\]
\[\text{ This will be the least when } r + \frac{2}{5} = 0, i . e . r = - \frac{2}{5} .\]
APPEARS IN
संबंधित प्रश्न
How many terms of G.P. 3, 32, 33, … are needed to give the sum 120?
If the 4th, 10th and 16th terms of a G.P. are x, y and z, respectively. Prove that x, y, z are in G.P.
Insert two numbers between 3 and 81 so that the resulting sequence is G.P.
The sum of some terms of G.P. is 315 whose first term and the common ratio are 5 and 2, respectively. Find the last term and the number of terms.
Find the 4th term from the end of the G.P.
Which term of the progression 18, −12, 8, ... is \[\frac{512}{729}\] ?
The seventh term of a G.P. is 8 times the fourth term and 5th term is 48. Find the G.P.
The sum of three numbers in G.P. is 21 and the sum of their squares is 189. Find the numbers.
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:
\[\frac{a}{1 + i} + \frac{a}{(1 + i )^2} + \frac{a}{(1 + i )^3} + . . . + \frac{a}{(1 + i )^n} .\]
Find the sum of the following geometric series:
x3, x5, x7, ... to n terms
Evaluate the following:
\[\sum^n_{k = 1} ( 2^k + 3^{k - 1} )\]
Evaluate the following:
\[\sum^{10}_{n = 2} 4^n\]
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: (21/4 . 41/8 . 81/16. 161/32 ... ∞) = 2.
If Sp denotes the sum of the series 1 + rp + r2p + ... to ∞ and sp the sum of the series 1 − rp + r2p − ... to ∞, prove that Sp + sp = 2 . S2p.
Find the rational numbers having the following decimal expansion:
\[0 . \overline3\]
If a, b, c are three distinct real numbers in G.P. and a + b + c = xb, then prove that either x< −1 or x > 3.
If pth, qth and rth terms of an A.P. and G.P. are both a, b and c respectively, show that \[a^{b - c} b^{c - a} c^{a - b} = 1\]
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)`.
If in an infinite G.P., first term is equal to 10 times the sum of all successive terms, then its common ratio is
The fractional value of 2.357 is
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.
The numbers x − 6, 2x and x2 are in G.P. Find x
The numbers x − 6, 2x and x2 are in G.P. Find nth term
For the following G.P.s, find Sn
3, 6, 12, 24, ...
Determine whether the sum to infinity of the following G.P.s exist, if exists find them:
`1/5, (-2)/5, 4/5, (-8)/5, 16/5, ...`
The sum of an infinite G.P. is 5 and the sum of the squares of these terms is 15 find the G.P.
A ball is dropped from a height of 10m. It bounces to a height of 6m, then 3.6m and so on. Find the total distance travelled by the ball
Find GM of two positive numbers whose A.M. and H.M. are 75 and 48
If a, b, c, d are four distinct positive quantities in G.P., then show that a + d > b + c
If pth, qth, and rth terms of an A.P. and G.P. are both a, b and c respectively, show that ab–c . bc – a . ca – b = 1
The third term of G.P. is 4. The product of its first 5 terms is ______.
The third term of a G.P. is 4, the product of the first five terms is ______.
