Advertisements
Advertisements
प्रश्न
Insert 5 geometric means between 16 and \[\frac{1}{4}\] .
Advertisements
उत्तर
\[\text { Let the 5 G . M . s betweem 16 and } \frac{1}{4} \text { be } G_1 , G_2 , G_3 , G_4 \text { and } G_5 . \]
\[16, G_1 , G_2 , G_3 , G_4 , G_5 , \frac{1}{4}\]
\[ \Rightarrow a = 16, n = 7 \text { and } a_7 = \frac{1}{4}\]
\[ \because a_7 = \frac{1}{4}\]
\[ \Rightarrow a r^6 = \frac{1}{4}\]
\[ \Rightarrow r^6 = \frac{1}{4 \times 16}\]
\[ \Rightarrow r^6 = \left( \frac{1}{2} \right)^6 \]
\[ \Rightarrow r = \frac{1}{2}\]
\[ \therefore G_1 = a_2 = ar = 16\left( \frac{1}{2} \right) = 8\]
\[ G_2 = a_3 = a r^2 = 16 \left( \frac{1}{2} \right)^2 = 4\]
\[ G_3 = a_4 = a r^3 = 16 \left( \frac{1}{2} \right)^3 = 2\]
\[ G_4 = a_5 = a r^4 = 16 \left( \frac{1}{2} \right)^4 = 1\]
\[ G_5 = a_6 = a r^5 = 16 \left( \frac{1}{2} \right)^5 = \frac{1}{2}\]
संबंधित प्रश्न
Which term of the following sequence:
`1/3, 1/9, 1/27`, ...., is `1/19683`?
The sum of first three terms of a G.P. is 16 and the sum of the next three terms is 128. Determine the first term, the common ratio and the sum to n terms of the G.P.
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
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 10th term of the G.P.
\[- \frac{3}{4}, \frac{1}{2}, - \frac{1}{3}, \frac{2}{9}, . . .\]
Which term of the G.P. :
\[\frac{1}{3}, \frac{1}{9}, \frac{1}{27} . . \text { . is } \frac{1}{19683} ?\]
The fourth term of a G.P. is 27 and the 7th term is 729, find the G.P.
Find the sum of the following geometric series:
0.15 + 0.015 + 0.0015 + ... to 8 terms;
Find the sum of the following geometric series:
\[\sqrt{2} + \frac{1}{\sqrt{2}} + \frac{1}{2\sqrt{2}} + . . .\text { to 8 terms };\]
How many terms of the series 2 + 6 + 18 + ... must be taken to make the sum equal to 728?
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.
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:
\[0 .\overline {231 }\]
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
\[\frac{2S S_1}{S^2 + S_1}\text { and } \frac{S^2 - S_1}{S^2 + S_1}\]
If a, b, c are in G.P., prove that:
\[\frac{1}{a^2 - b^2} + \frac{1}{b^2} = \frac{1}{b^2 - c^2}\]
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, c are in A.P. and a, b, d are in G.P., then prove that a, a − b, d − c are in G.P.
If a, b, c are in A.P. and a, x, b and b, y, c are in G.P., show that x2, b2, y2 are in A.P.
The nth term of a G.P. is 128 and the sum of its n terms is 255. If its common ratio is 2, then its first term is ______.
If a, b, c are in G.P. and x, y are AM's between a, b and b,c respectively, then
If A be one A.M. and p, q be two G.M.'s between two numbers, then 2 A is equal to
The product (32), (32)1/6 (32)1/36 ... to ∞ is equal to
The two geometric means between the numbers 1 and 64 are
For the G.P. if a = `7/243`, r = 3 find t6.
The fifth term of a G.P. is x, eighth term of a G.P. is y and eleventh term of a G.P. is z verify whether y2 = xz
A ball is dropped from a height of 80 ft. The ball is such that it rebounds `(3/4)^"th"` of the height it has fallen. How high does the ball rebound on 6th bounce? How high does the ball rebound on nth bounce?
Determine whether the sum to infinity of the following G.P.s exist, if exists find them:
`2, 4/3, 8/9, 16/27, ...`
Express the following recurring decimal as a rational number:
`2.3bar(5)`
Select the correct answer from the given alternative.
If for a G.P. `"t"_6/"t"_3 = 1458/54` then r = ?
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:
For a sequence Sn = 4(7n – 1) verify that the sequence is a G.P.
Answer the following:
For a G.P. if t2 = 7, t4 = 1575 find a
Answer the following:
If p, q, r, s are in G.P., show that (p2 + q2 + r2) (q2 + r2 + s2) = (pq + qr + rs)2
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 ______.
If the expansion in powers of x of the function `1/((1 - ax)(1 - bx))` is a0 + a1x + a2x2 + a3x3 ....... then an is ______.
