Advertisements
Advertisements
प्रश्न
Find three numbers in G.P. whose product is 729 and the sum of their products in pairs is 819.
Advertisements
उत्तर
Let the required numbers be \[\frac{a}{r}, \text { a and ar } .\]
Product of the G.P. = 729
\[\Rightarrow a^3 = 729\]
\[ \Rightarrow a = 9\]
Sum of the products in pairs = 819
\[\Rightarrow \frac{a}{r} \times a + a \times ar + ar \times \frac{a}{r} = 819\]
\[ \Rightarrow a^2 \left( \frac{1}{r} + r + 1 \right) = 819\]
\[ \Rightarrow 81\left( \frac{1 + r^2 + r}{r} \right) = 819\]
\[ \Rightarrow 9\left( r^2 + r + 1 \right) = 91r\]
\[ \Rightarrow 9 r^2 - 82r + 9 = 0\]
\[ \Rightarrow 9 r^2 - 81r - r + 9 = 0\]
\[ \Rightarrow \left( 9r - 1 \right)\left( r - 9 \right) = 0\]
\[ \Rightarrow r = \frac{1}{9}, 9\]
\[\text { Hence, putting the values of a and r, we get the numbers to be 81, 9 and 1 or 1, 9 and 81 } .\]
APPEARS IN
संबंधित प्रश्न
Which term of the following sequence:
`2, 2sqrt2, 4,.... is 128`
Which term of the following sequence:
`1/3, 1/9, 1/27`, ...., is `1/19683`?
Given a G.P. with a = 729 and 7th term 64, determine S7.
Find a G.P. for which sum of the first two terms is –4 and the fifth term is 4 times the third term.
If the first and the nth term of a G.P. are a ad b, respectively, and if P is the product of n terms, prove that P2 = (ab)n.
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`.
The first term of a G.P. is 1. The sum of the third term and fifth term is 90. Find the common ratio of G.P.
Which term of the progression 0.004, 0.02, 0.1, ... is 12.5?
If the G.P.'s 5, 10, 20, ... and 1280, 640, 320, ... have their nth terms equal, find the value of n.
The 4th term of a G.P. is square of its second term, and the first term is − 3. Find its 7th term.
Find three numbers in G.P. whose sum is 65 and whose product is 3375.
The sum of three numbers in G.P. is 14. If the first two terms are each increased by 1 and the third term decreased by 1, the resulting numbers are in A.P. Find the numbers.
Evaluate the following:
\[\sum^{10}_{n = 2} 4^n\]
How many terms of the sequence \[\sqrt{3}, 3, 3\sqrt{3},\] ... must be taken to make the sum \[39 + 13\sqrt{3}\] ?
If S1, S2, ..., Sn are the sums of n terms of n G.P.'s whose first term is 1 in each and common ratios are 1, 2, 3, ..., n respectively, then prove that S1 + S2 + 2S3 + 3S4 + ... (n − 1) Sn = 1n + 2n + 3n + ... + nn.
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.
Express the recurring decimal 0.125125125 ... as a rational number.
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.
The sum of three numbers in G.P. is 56. If we subtract 1, 7, 21 from these numbers in that order, we obtain an A.P. Find the numbers.
If a, b, c are in G.P., prove that:
\[\frac{(a + b + c )^2}{a^2 + b^2 + c^2} = \frac{a + b + c}{a - b + c}\]
If a, b, c are in G.P., prove that the following is also in G.P.:
a2 + b2, ab + bc, b2 + c2
Insert 5 geometric means between 16 and \[\frac{1}{4}\] .
Find the geometric means of the following pairs of number:
2 and 8
If the sum of an infinite decreasing G.P. is 3 and the sum of the squares of its term is \[\frac{9}{2}\], then write its first term and common difference.
Write the product of n geometric means between two numbers a and b.
If in an infinite G.P., first term is equal to 10 times the sum of all successive terms, then its common ratio is
If p, q be two A.M.'s and G be one G.M. between two numbers, then G2 =
For the G.P. if r = `1/3`, a = 9 find t7
For the G.P. if a = `7/243`, r = 3 find t6.
Find three numbers in G.P. such that their sum is 21 and sum of their squares is 189.
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?
For a G.P. If t4 = 16, t9 = 512, find S10
The midpoints of the sides of a square of side 1 are joined to form a new square. This procedure is repeated indefinitely. Find the sum of the perimeters of all the squares
Answer the following:
Find the sum of the first 5 terms of the G.P. whose first term is 1 and common ratio is `2/3`
Answer the following:
Find k so that k – 1, k, k + 2 are consecutive terms of a G.P.
If a, b, c, d are in G.P., prove that a2 – b2, b2 – c2, c2 – d2 are also in G.P.
In a G.P. of positive terms, if any term is equal to the sum of the next two terms. Then the common ratio of the G.P. is ______.
