Advertisements
Advertisements
प्रश्न
Insert 5 geometric means between 16 and \[\frac{1}{4}\] .
Advertisements
उत्तर
\[\text { Let the 5 G . M . s betweem 16 and } \frac{1}{4} \text { be } G_1 , G_2 , G_3 , G_4 \text { and } G_5 . \]
\[16, G_1 , G_2 , G_3 , G_4 , G_5 , \frac{1}{4}\]
\[ \Rightarrow a = 16, n = 7 \text { and } a_7 = \frac{1}{4}\]
\[ \because a_7 = \frac{1}{4}\]
\[ \Rightarrow a r^6 = \frac{1}{4}\]
\[ \Rightarrow r^6 = \frac{1}{4 \times 16}\]
\[ \Rightarrow r^6 = \left( \frac{1}{2} \right)^6 \]
\[ \Rightarrow r = \frac{1}{2}\]
\[ \therefore G_1 = a_2 = ar = 16\left( \frac{1}{2} \right) = 8\]
\[ G_2 = a_3 = a r^2 = 16 \left( \frac{1}{2} \right)^2 = 4\]
\[ G_3 = a_4 = a r^3 = 16 \left( \frac{1}{2} \right)^3 = 2\]
\[ G_4 = a_5 = a r^4 = 16 \left( \frac{1}{2} \right)^4 = 1\]
\[ G_5 = a_6 = a r^5 = 16 \left( \frac{1}{2} \right)^5 = \frac{1}{2}\]
APPEARS IN
संबंधित प्रश्न
Find the 20th and nthterms of the G.P. `5/2, 5/4 , 5/8,...`
Find the sum to 20 terms in the geometric progression 0.15, 0.015, 0.0015,…
Given a G.P. with a = 729 and 7th term 64, determine S7.
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
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.
Which term of the progression 18, −12, 8, ... is \[\frac{512}{729}\] ?
Find the 4th term from the end of the G.P.
\[\frac{1}{2}, \frac{1}{6}, \frac{1}{18}, \frac{1}{54}, . . . , \frac{1}{4374}\]
If the G.P.'s 5, 10, 20, ... and 1280, 640, 320, ... have their nth terms equal, find the value of n.
Find the sum of the following geometric progression:
1, 3, 9, 27, ... to 8 terms;
Find the sum of the following geometric series:
0.15 + 0.015 + 0.0015 + ... to 8 terms;
The sum of n terms of the G.P. 3, 6, 12, ... is 381. Find the value of n.
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.
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 rational numbers having the following decimal expansion:
\[3 . 5\overline 2\]
One side of an equilateral triangle is 18 cm. The mid-points of its sides are joined to form another triangle whose mid-points, in turn, are joined to form still another triangle. The process is continued indefinitely. Find the sum of the (i) perimeters of all the triangles. (ii) areas of all triangles.
The sum of first two terms of an infinite G.P. is 5 and each term is three times the sum of the succeeding terms. Find the G.P.
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), (b − c), (c − a) are in G.P., then prove that (a + b + c)2 = 3 (ab + bc + ca)
Insert 5 geometric means between \[\frac{32}{9}\text{and}\frac{81}{2}\] .
Find the geometric means of the following pairs of number:
−8 and −2
If in an infinite G.P., first term is equal to 10 times the sum of all successive terms, then its common ratio is
Given that x > 0, the sum \[\sum^\infty_{n = 1} \left( \frac{x}{x + 1} \right)^{n - 1}\] equals
Find four numbers in G.P. such that sum of the middle two numbers is `10/3` and their product is 1
For the following G.P.s, find Sn.
`sqrt(5)`, −5, `5sqrt(5)`, −25, ...
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.
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]
Determine whether the sum to infinity of the following G.P.s exist, if exists find them:
`-3, 1, (-1)/3, 1/9, ...`
Select the correct answer from the given alternative.
Which term of the geometric progression 1, 2, 4, 8, ... is 2048
Select the correct answer from the given alternative.
Which of the following is not true, where A, G, H are the AM, GM, HM of a and b respectively. (a, b > 0)
Answer the following:
Find three numbers in G.P. such that their sum is 35 and their product is 1000
Answer the following:
If p, q, r, s are in G.P., show that (pn + qn), (qn + rn) , (rn + sn) are also in G.P.
In a G.P. of even number of terms, the sum of all terms is 5 times the sum of the odd terms. The common ratio of the G.P. 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 ______.
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 ______.
If in a geometric progression {an}, a1 = 3, an = 96 and Sn = 189, then the value of n is ______.
