Advertisements
Advertisements
Question
Insert 5 geometric means between 16 and \[\frac{1}{4}\] .
Advertisements
Solution
\[\text { Let the 5 G . M . s betweem 16 and } \frac{1}{4} \text { be } G_1 , G_2 , G_3 , G_4 \text { and } G_5 . \]
\[16, G_1 , G_2 , G_3 , G_4 , G_5 , \frac{1}{4}\]
\[ \Rightarrow a = 16, n = 7 \text { and } a_7 = \frac{1}{4}\]
\[ \because a_7 = \frac{1}{4}\]
\[ \Rightarrow a r^6 = \frac{1}{4}\]
\[ \Rightarrow r^6 = \frac{1}{4 \times 16}\]
\[ \Rightarrow r^6 = \left( \frac{1}{2} \right)^6 \]
\[ \Rightarrow r = \frac{1}{2}\]
\[ \therefore G_1 = a_2 = ar = 16\left( \frac{1}{2} \right) = 8\]
\[ G_2 = a_3 = a r^2 = 16 \left( \frac{1}{2} \right)^2 = 4\]
\[ G_3 = a_4 = a r^3 = 16 \left( \frac{1}{2} \right)^3 = 2\]
\[ G_4 = a_5 = a r^4 = 16 \left( \frac{1}{2} \right)^4 = 1\]
\[ G_5 = a_6 = a r^5 = 16 \left( \frac{1}{2} \right)^5 = \frac{1}{2}\]
RELATED QUESTIONS
Which term of the following sequence:
`2, 2sqrt2, 4,.... is 128`
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.
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.
Show that the ratio of the sum of first n terms of a G.P. to the sum of terms from (n + 1)th to (2n)th term is `1/r^n`.
Find :
the 12th term of the G.P.
\[\frac{1}{a^3 x^3}, ax, a^5 x^5 , . . .\]
If the G.P.'s 5, 10, 20, ... and 1280, 640, 320, ... have their nth terms equal, find the value of n.
The product of three numbers in G.P. is 216. If 2, 8, 6 be added to them, the results are in A.P. Find the numbers.
Find the sum of the following geometric progression:
1, 3, 9, 27, ... to 8 terms;
Find the sum of the following geometric series:
\[\frac{a}{1 + i} + \frac{a}{(1 + i )^2} + \frac{a}{(1 + i )^3} + . . . + \frac{a}{(1 + i )^n} .\]
How many terms of the sequence \[\sqrt{3}, 3, 3\sqrt{3},\] ... must be taken to make the sum \[39 + 13\sqrt{3}\] ?
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.
Find the sum of 2n terms of the series whose every even term is 'a' times the term before it and every odd term is 'c' times the term before it, the first term being unity.
Prove that: (21/4 . 41/8 . 81/16. 161/32 ... ∞) = 2.
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), (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:
Insert 6 geometric means between 27 and \[\frac{1}{81}\] .
Find the geometric means of the following pairs of number:
2 and 8
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
The fractional value of 2.357 is
The sum of an infinite G.P. is 4 and the sum of the cubes of its terms is 92. The common ratio of the original G.P. 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
For the G.P. if a = `7/243`, r = 3 find t6.
For the G.P. if a = `2/3`, t6 = 162, find r.
Find five numbers in G.P. such that their product is 1024 and fifth term is square of the third 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 3 years.
For a G.P. a = 2, r = `-2/3`, find S6
If the common ratio of a G.P. is `2/3` and sum to infinity is 12. Find the first term
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
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 -
The sum of 3 terms of a G.P. is `21/4` and their product is 1 then the common ratio is ______.
Answer the following:
In a G.P., the fourth term is 48 and the eighth term is 768. Find the tenth term
Answer the following:
Find three numbers in G.P. such that their sum is 35 and their product is 1000
Answer the following:
If a, b, c are in G.P. and ax2 + 2bx + c = 0 and px2 + 2qx + r = 0 have common roots then verify that pb2 – 2qba + ra2 = 0
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
For a, b, c to be in G.P. the value of `(a - b)/(b - c)` is equal to ______.
