Advertisements
Advertisements
प्रश्न
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.
Advertisements
उत्तर
Let the terms of the given G.P. be \[\frac{a}{r}, \text { a and ar }\]
∴ Product = 216
\[\Rightarrow a^3 = 216\]
\[ \Rightarrow a = 6\]
It is given that \[\frac{a}{r} + 2, a + 8 \text { and ar } + 6\] are in A.P.
\[\therefore 2\left( a + 8 \right) = \frac{a}{r} + 2 + ar + 6\]
\[\text { Putting a = 6, we get }\]
\[ \Rightarrow 28 = \frac{6}{r} + 2 + 6r + 6\]
\[ \Rightarrow 28r = 6 r^2 + 8r + 6\]
\[ \Rightarrow 6 r^2 - 20r + 6 = 0\]
\[ \Rightarrow \left( 6r - 2 \right)\left( r - 3 \right) = 0\]
\[ \Rightarrow r = \frac{1}{3}, 3\]
\[\text { Hence, putting the values of a and r, the required numbers are 18, 6, 2 or 2, 6 and 18 }.\]
संबंधित प्रश्न
Find the sum to n terms of the sequence, 8, 88, 888, 8888… .
If the pth, qth and rth terms of a G.P. are a, b and c, respectively. Prove that `a^(q - r) b^(r-p) c^(p-q) = 1`.
The sum of two numbers is 6 times their geometric mean, show that numbers are in the ratio `(3 + 2sqrt2) ":" (3 - 2sqrt2)`.
Show that one of the following progression is a G.P. Also, find the common ratio in case:
−2/3, −6, −54, ...
Show that one of the following progression is a G.P. Also, find the common ratio in case:1/2, 1/3, 2/9, 4/27, ...
The fourth term of a G.P. is 27 and the 7th term is 729, find the G.P.
If a, b, c, d and p are different real numbers such that:
(a2 + b2 + c2) p2 − 2 (ab + bc + cd) p + (b2 + c2 + d2) ≤ 0, then show that a, b, c and d are in 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:
\[\sqrt{2} + \frac{1}{\sqrt{2}} + \frac{1}{2\sqrt{2}} + . . .\text { to 8 terms };\]
Find the sum of the following geometric series:
1, −a, a2, −a3, ....to n terms (a ≠ 1)
Find the sum of the following series:
0.5 + 0.55 + 0.555 + ... to n terms.
How many terms of the G.P. 3, 3/2, 3/4, ... be taken together to make \[\frac{3069}{512}\] ?
How many terms of the sequence \[\sqrt{3}, 3, 3\sqrt{3},\] ... must be taken to make the sum \[39 + 13\sqrt{3}\] ?
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.
If a and b are the roots of x2 − 3x + p = 0 and c, d are the roots x2 − 12x + q = 0, where a, b, c, d form a G.P. Prove that (q + p) : (q − p) = 17 : 15.
A person has 2 parents, 4 grandparents, 8 great grandparents, and so on. Find the number of his ancestors during the ten generations preceding his own.
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 terms of an infinite decreasing G.P. in which all the terms are positive, the first term is 4, and the difference between the third and fifth term is equal to 32/81.
Find k such that k + 9, k − 6 and 4 form three consecutive terms of a 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, d are in G.P., prove that:
(a2 − b2), (b2 − c2), (c2 − d2) 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)
If \[\frac{1}{a + b}, \frac{1}{2b}, \frac{1}{b + c}\] are three consecutive terms of an A.P., prove that a, b, c are the three consecutive terms of a G.P.
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:
a3b and ab3
The two geometric means between the numbers 1 and 64 are
In a G.P. if the (m + n)th term is p and (m − n)th term is q, then its mth term is
Mark the correct alternative in the following question:
Let S be the sum, P be the product and R be the sum of the reciprocals of 3 terms of a G.P. Then p2R3 : S3 is equal to
Check whether the following sequence is G.P. If so, write tn.
`sqrt(5), 1/sqrt(5), 1/(5sqrt(5)), 1/(25sqrt(5))`, ...
Check whether the following sequence is G.P. If so, write tn.
3, 4, 5, 6, …
Find three numbers in G.P. such that their sum is 21 and sum of their squares is 189.
Find five numbers in G.P. such that their product is 1024 and fifth term is square of the third term.
If p, q, r, s are in G.P. show that p + q, q + r, r + s are also in G.P.
The numbers 3, x, and x + 6 form are in G.P. Find 20th 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 10 years.
The numbers x − 6, 2x and x2 are in G.P. Find 1st term
For the following G.P.s, find Sn.
`sqrt(5)`, −5, `5sqrt(5)`, −25, ...
Let S be the sum, P be the product and R be the sum of the reciprocals of 3 terms of a G.P. Then P2 R3 : S3 is equal to ______.
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 ______.
If in a geometric progression {an}, a1 = 3, an = 96 and Sn = 189, then the value of n is ______.
