Advertisements
Advertisements
Questions
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.
The sum of three consecutive numbers of a G.P. is 14. If 1 is added in first and second term each and 1 subtracted from third, the new numbers form an A.P. Find the numbers.
Advertisements
Solution
Let the numbers be a, ar and ar2.
\[\text { Sum }= 14 \]
\[ \Rightarrow a + ar + a r^2 = 14 \]
\[ \Rightarrow a(1 + r + r^2 ) = 14 . . . \left( i \right)\]
According to the question,a + 1, ar + 1 and ar2 − 1 are in A.P.
\[\therefore 2\left( ar + 1 \right) = a + 1 + a r^2 - 1\]
\[ \Rightarrow 2ar + 2 = a + a r^2 \]
\[ \Rightarrow 2ar + 2 = 14 - ar [\text { From }\left( i \right)]\]
\[ \Rightarrow 3ar = 12 \]
\[ \Rightarrow a = \frac{4}{r} . . . \left( ii \right)\]
\[\text { Putting } a = \frac{4}{r} \text { in }\left( i \right)\]
\[ \Rightarrow \frac{4}{r}(1 + r + r^2 ) = 14\]
\[ \Rightarrow 4 r^2 - 10r + 4 = 0 \]
\[ \Rightarrow 4 r^2 - 8r - 2r + 4 = 0 \]
\[ \Rightarrow \left( 4r - 2 \right)\left( r - 2 \right) = 0\]
\[ \Rightarrow r = \frac{1}{2}, 2\]
\[\text { Putting } r = \frac{1}{2}\text { in } \left( ii \right), \text { we get a } = 8 . \]
\[\text { So, the G . P . is 8, 4 and } 2 . \]
Similarly putting r = 2 in (ii), we get a = 2.
So, the G.P is 2, 4 and 8.
APPEARS IN
RELATED QUESTIONS
The 5th, 8th and 11th terms of a G.P. are p, q and s, respectively. Show that q2 = ps.
Find the sum to n terms of the sequence, 8, 88, 888, 8888… .
If a, b, c, d are in G.P, prove that (an + bn), (bn + cn), (cn + dn) are in G.P.
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:
\[a, \frac{3 a^2}{4}, \frac{9 a^3}{16}, . . .\]
Find :
nth term of the G.P.
\[\sqrt{3}, \frac{1}{\sqrt{3}}, \frac{1}{3\sqrt{3}}, . . .\]
The fourth term of a G.P. is 27 and the 7th term is 729, find the G.P.
The 4th term of a G.P. is square of its second term, and the first term is − 3. Find its 7th term.
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.
Find the sum of the following geometric progression:
1, 3, 9, 27, ... to 8 terms;
Find the sum of the following geometric progression:
(a2 − b2), (a − b), \[\left( \frac{a - b}{a + b} \right)\] to n terms;
Find the sum of the following series:
9 + 99 + 999 + ... to n terms;
The sum of n terms of the G.P. 3, 6, 12, ... is 381. Find the value of n.
Find the sum :
\[\sum^{10}_{n = 1} \left[ \left( \frac{1}{2} \right)^{n - 1} + \left( \frac{1}{5} \right)^{n + 1} \right] .\]
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 \[\frac{1}{r^n}\].
Find k such that k + 9, k − 6 and 4 form three consecutive terms of a G.P.
If a, b, c, d are in G.P., prove that:
(a + b + c + d)2 = (a + b)2 + 2 (b + c)2 + (c + d)2
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.
The sum of two numbers is 6 times their geometric means, show that the numbers are in the ratio `(3+2sqrt2):(3-2sqrt2)`.
Given that x > 0, the sum \[\sum^\infty_{n = 1} \left( \frac{x}{x + 1} \right)^{n - 1}\] equals
For the G.P. if a = `7/243`, r = 3 find t6.
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 x
The numbers 3, x, and x + 6 form are in G.P. Find nth 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 n years.
If S, P, R are the sum, product, and sum of the reciprocals of n terms of a G.P. respectively, then verify that `["S"/"R"]^"n"` = P2
Find: `sum_("r" = 1)^10(3 xx 2^"r")`
Find GM of two positive numbers whose A.M. and H.M. are 75 and 48
Insert two numbers between 1 and −27 so that the resulting sequence is a G.P.
Select the correct answer from the given alternative.
The tenth term of the geometric sequence `1/4, (-1)/2, 1, -2,` ... is –
Select the correct answer from the given alternative.
Sum to infinity of a G.P. 5, `-5/2, 5/4, -5/8, 5/16,...` is –
Answer the following:
For a G.P. a = `4/3` and t7 = `243/1024`, find the value of r
If the pth and qth terms of a G.P. are q and p respectively, show that its (p + q)th term is `(q^p/p^q)^(1/(p - q))`
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.
If `e^((cos^2x + cos^4x + cos^6x + ...∞)log_e2` satisfies the equation t2 – 9t + 8 = 0, then the value of `(2sinx)/(sinx + sqrt(3)cosx)(0 < x ,< π/2)` is ______.
