Advertisements
Advertisements
प्रश्न
Find four numbers in G.P. such that sum of the middle two numbers is `10/3` and their product is 1
Advertisements
उत्तर
Let the four numbers in G.P. be `"a"/"r"^3, "a"/"r", "ar", "ar"^3`
Since their product is 1, `"a"/"r"^3*"a"/"r"*"ar"*"ar"^3` = 1
∴ a4 = 1
∴ a = 1
Also the sum of middle two numbers is `10/3`
∴ `"a"/"r" + "ar" = 10/3`
∴ `"a"(1/"r" + "r") = 10/3`
∴ `1/"r" + "r" = 10/3` as a = 1
∴ `(1 + "r"^2)/"r" = 10/3`
∴ 3 + 3r2 = 10r
∴ 3r2 – 10r + 3 = 0
∴ (r – 3)(3r – 1) = 0
∴ r = 3 or r = `1/3`
Taking r = 3, `"a"/"r"^3 = 1/27, "a"/"r" = 1/3`, ar3 = 27 and the four numbers are `1/27, 1/3, 3, 27`.
Taking r = `1/3`, `"a"/"r"^3 = 1/((1/27))` = 27, `"a"/"r" = 1/((1/3))` = 3, `"ar" = 1/3`, ar3 = `1/27` and the our numbers are 27, 3, `1/3`, `1/27`.
Hence, the required numbers in G.P. are `1/27, 1/3, 3, 27` or `27, 3, 1/3, 1/27`.
APPEARS IN
संबंधित प्रश्न
Find the 20th and nthterms of the G.P. `5/2, 5/4 , 5/8,...`
Which term of the following sequence:
`1/3, 1/9, 1/27`, ...., is `1/19683`?
Insert two numbers between 3 and 81 so that the resulting sequence is G.P.
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, ...
Find the 4th term from the end of the G.P.
\[\frac{1}{2}, \frac{1}{6}, \frac{1}{18}, \frac{1}{54}, . . . , \frac{1}{4374}\]
The sum of first three terms of a G.P. is 13/12 and their product is − 1. Find the G.P.
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;
How many terms of the G.P. 3, 3/2, 3/4, ... be taken together to make \[\frac{3069}{512}\] ?
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 series to infinity:
`1/3+1/5^2 +1/3^3+1/5^4 + 1/3^5 + 1/56+ ...infty`
Express the recurring decimal 0.125125125 ... as a rational number.
Find the rational numbers having the following decimal expansion:
\[3 . 5\overline 2\]
Find an infinite G.P. whose first term is 1 and each term is the sum of all the terms which follow it.
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 are in G.P., prove that the following is also in G.P.:
a2 + b2, ab + bc, b2 + c2
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 the fifth term of a G.P. is 2, then write the product of its 9 terms.
If A1, A2 be two AM's and G1, G2 be two GM's between a and b, then find the value of \[\frac{A_1 + A_2}{G_1 G_2}\]
Write the product of n geometric means between two numbers a and b.
If the first term of a G.P. a1, a2, a3, ... is unity such that 4 a2 + 5 a3 is least, then the common ratio of G.P. is
If second term of a G.P. is 2 and the sum of its infinite terms is 8, then its first term is
Given that x > 0, the sum \[\sum^\infty_{n = 1} \left( \frac{x}{x + 1} \right)^{n - 1}\] equals
In a G.P. of even number of terms, the sum of all terms is five times the sum of the odd terms. The common ratio of the G.P. is
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
Check whether the following sequence is G.P. If so, write tn.
7, 14, 21, 28, …
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 x
For a G.P. if a = 2, r = 3, Sn = 242 find n
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
Determine whether the sum to infinity of the following G.P.s exist, if exists find them:
`-3, 1, (-1)/3, 1/9, ...`
Find : `sum_("r" = 1)^oo 4(0.5)^"r"`
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`
The third term of G.P. is 4. The product of its first 5 terms is ______.
If x, 2y, 3z are in A.P., where the distinct numbers x, y, z are in G.P. then the common ratio of the G.P. is ______.
