Advertisements
Advertisements
प्रश्न
Answer the following:
Find five numbers in G.P. such that their product is 243 and sum of second and fourth number is 10.
उत्तर
Let the five numbers in G.P. be `"a"/"r"^2, "a"/"r","a", "ar", "ar"^2`
Since their product is 243,
`"a"/"r"^2."a"/r."a"."ar"."ar"^2`= 243
∴ a5 = 35
∴ a = 3
Also, the sum of second and fourth is 10
∴ `"a"/"r" + "ar"` = 10
∴ `3/"r" + 3"r"` = 10r ...[∵ a = 3]
∴ 3 + 3r2 = 10r
∴ 3r2 − 10r + 3 = 0
∴ (r – 3)(3r – 1) = 0
∴ r – 3 = 0 or 3r – 1= 0
∴ r = 3 or r = `1/3`
Taking r = `3, "a"/"r"^2 = 3/9 = 1/3, "a"/"r" = 3/3` = 1, ar = 3 × 3 = 9,
ar2 = 3(3)2 = 27 and the five numbers are `1/3`, 1, 3, 9, 27
Taking r = `1/3, "a"/"r"^2 = 3/((1/9)) = 27, "a"/"r" = 3/((1/3))` = 9,
ar = `3(1/3)` = 1, ar2 = `3(1/3)^2 = 1/3`
and the five numbers are 27, 9, 3, 1, `1/3`
Hence, the required numbers in G.P. are `1/3`, 1, 3, 9, 27 or 27, 9, 3, 1, `1/3`.
APPEARS IN
संबंधित प्रश्न
Find the sum to 20 terms in the geometric progression 0.15, 0.015, 0.0015,…
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`
Insert two numbers between 3 and 81 so that the resulting sequence is G.P.
Which term of the progression 0.004, 0.02, 0.1, ... is 12.5?
Which term of the G.P. :
\[\sqrt{2}, \frac{1}{\sqrt{2}}, \frac{1}{2\sqrt{2}}, \frac{1}{4\sqrt{2}}, . . . \text { is }\frac{1}{512\sqrt{2}}?\]
The seventh term of a G.P. is 8 times the fourth term and 5th term is 48. Find the G.P.
If \[\frac{a + bx}{a - bx} = \frac{b + cx}{b - cx} = \frac{c + dx}{c - dx}\] (x ≠ 0), then show that a, b, c and d are in G.P.
Find the sum of the following geometric progression:
4, 2, 1, 1/2 ... to 10 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:
x3, x5, x7, ... to n terms
If pth, qth, rth and sth terms of an A.P. be in G.P., then prove that p − q, q − r, r − s 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.
If a, b, c are in A.P. and a, b, d are in G.P., show that a, (a − b), (d − c) are in G.P.
Find the geometric means of the following pairs of number:
−8 and −2
If pth, qth and rth terms of a G.P. re x, y, z respectively, then write the value of xq − r yr − pzp − q.
The fractional value of 2.357 is
If the sum of first two terms of an infinite GP is 1 every term is twice the sum of all the successive terms, then its first term is
If second term of a G.P. is 2 and the sum of its infinite terms is 8, 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
Given that x > 0, the sum \[\sum^\infty_{n = 1} \left( \frac{x}{x + 1} \right)^{n - 1}\] equals
Let x be the A.M. and y, z be two G.M.s between two positive numbers. Then, \[\frac{y^3 + z^3}{xyz}\] is equal to
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))`, ...
Find three numbers in G.P. such that their sum is 21 and sum of their squares is 189.
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 3 years.
The numbers x − 6, 2x and x2 are in G.P. Find x
For the following G.P.s, find Sn.
`sqrt(5)`, −5, `5sqrt(5)`, −25, ...
For a G.P. If t4 = 16, t9 = 512, find S10
Find the sum to n terms of the sequence.
0.5, 0.05, 0.005, ...
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
If one invests Rs. 10,000 in a bank at a rate of interest 8% per annum, how long does it take to double the money by compound interest? [(1.08)5 = 1.47]
Answer the following:
For a sequence , if tn = `(5^("n" - 2))/(7^("n" - 3))`, verify whether the sequence is a G.P. If it is a G.P., find its first term and the common ratio.
Answer the following:
If for a G.P. first term is (27)2 and seventh term is (8)2, find S8
The third term of G.P. is 4. The product of its first 5 terms 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 ______.
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 ______.
If the expansion in powers of x of the function `1/((1 - ax)(1 - bx))` is a0 + a1x + a2x2 + a3x3 ....... then an is ______.
