Advertisements
Advertisements
प्रश्न
The two geometric means between the numbers 1 and 64 are
विकल्प
(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
How many terms of G.P. 3, 32, 33, … are needed to give the sum 120?
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 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 4th term from the end of the G.P.
Which term of the G.P. :
\[\sqrt{2}, \frac{1}{\sqrt{2}}, \frac{1}{2\sqrt{2}}, \frac{1}{4\sqrt{2}}, . . . \text { is }\frac{1}{512\sqrt{2}}?\]
The fourth term of a G.P. is 27 and the 7th term is 729, find the G.P.
The product of three numbers in G.P. is 125 and the sum of their products taken in pairs is \[87\frac{1}{2}\] . Find them.
Find the sum of the following geometric series:
(x +y) + (x2 + xy + y2) + (x3 + x2y + xy2 + y3) + ... to n terms;
Find the sum of the following series:
0.6 + 0.66 + 0.666 + .... to n terms
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 the first three terms to that of the first 6 terms of a G.P. is 125 : 152. Find the common ratio.
The 4th and 7th terms of a G.P. are \[\frac{1}{27} \text { and } \frac{1}{729}\] respectively. Find the sum of n terms of the G.P.
Let an be the nth term of the G.P. of positive numbers.
Let \[\sum^{100}_{n = 1} a_{2n} = \alpha \text { and } \sum^{100}_{n = 1} a_{2n - 1} = \beta,\] such that α ≠ β. Prove that the common ratio of the G.P. is α/β.
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 serie to infinity:
`2/5 + 3/5^2 +2/5^3 + 3/5^4 + ... ∞.`
If a, b, c are in G.P., prove that \[\frac{1}{\log_a m}, \frac{1}{\log_b m}, \frac{1}{\log_c m}\] are in A.P.
The sum of three numbers a, b, c in A.P. is 18. If a and b are each increased by 4 and c is increased by 36, the new numbers form a G.P. Find a, b, c.
If pth, qth and rth terms of an A.P. and G.P. are both a, b and c respectively, show that \[a^{b - c} b^{c - a} c^{a - b} = 1\]
Find the geometric means of the following pairs of number:
−8 and −2
If pth, qth and rth terms of an A.P. are in G.P., then the common ratio of this G.P. is
If a, b, c are in G.P. and x, y are AM's between a, b and b,c respectively, then
Given that x > 0, the sum \[\sum^\infty_{n = 1} \left( \frac{x}{x + 1} \right)^{n - 1}\] equals
In a G.P. if the (m + n)th term is p and (m − n)th term is q, then its mth term is
Find five numbers in G.P. such that their product is 1024 and fifth term is square of the third term.
The numbers x − 6, 2x and x2 are in G.P. Find nth term
For the following G.P.s, find Sn.
`sqrt(5)`, −5, `5sqrt(5)`, −25, ...
For a G.P. If t3 = 20 , t6 = 160 , find S7
The value of a house appreciates 5% per year. How much is the house worth after 6 years if its current worth is ₹ 15 Lac. [Given: (1.05)5 = 1.28, (1.05)6 = 1.34]
Express the following recurring decimal as a rational number:
`51.0bar(2)`
The sum of an infinite G.P. is 5 and the sum of the squares of these terms is 15 find the G.P.
Find : `sum_("r" = 1)^oo (-1/3)^"r"`
Answer the following:
For a G.P. if t2 = 7, t4 = 1575 find a
Answer the following:
Find `sum_("r" = 1)^"n" (2/3)^"r"`
Answer the following:
If for a G.P. first term is (27)2 and seventh term is (8)2, find S8
The third term of G.P. is 4. The product of its first 5 terms is ______.
The lengths of three unequal edges of a rectangular solid block are in G.P. The volume of the block is 216 cm3 and the total surface area is 252cm2. The length of the longest edge is ______.
For a, b, c to be in G.P. the value of `(a - b)/(b - c)` is equal to ______.
Find a G.P. for which sum of the first two terms is – 4 and the fifth term is 4 times the third term.
The sum of the first three terms of a G.P. is S and their product is 27. Then all such S lie in ______.
