Advertisements
Advertisements
Question
Find four numbers in G.P. such that sum of the middle two numbers is `10/3` and their product is 1
Advertisements
Solution
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
RELATED QUESTIONS
Find the 20th and nthterms of the G.P. `5/2, 5/4 , 5/8,...`
The 4th term of a G.P. is square of its second term, and the first term is –3. Determine its 7thterm.
Which term of the following sequence:
`sqrt3, 3, 3sqrt3`, .... is 729?
Find four numbers forming a geometric progression in which third term is greater than the first term by 9, and the second term is greater than the 4th by 18.
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`.
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.
The sum of two numbers is 6 times their geometric mean, show that numbers are in the ratio `(3 + 2sqrt2) ":" (3 - 2sqrt2)`.
If f is a function satisfying f (x +y) = f(x) f(y) for all x, y ∈ N such that f(1) = 3 and `sum_(x = 1)^n` f(x) = 120, find the value of n.
A G.P. consists of an even number of terms. If the sum of all the terms is 5 times the sum of terms occupying odd places, then find its common ratio.
Show that the sequence <an>, defined by an = \[\frac{2}{3^n}\], n ϵ N is a G.P.
Find the 4th term from the end of the G.P.
\[\frac{1}{2}, \frac{1}{6}, \frac{1}{18}, \frac{1}{54}, . . . , \frac{1}{4374}\]
If the pth and qth terms of a G.P. are q and p, respectively, then show that (p + q)th term is \[\left( \frac{q^p}{p^q} \right)^\frac{1}{p - q}\].
Find the sum of the following geometric progression:
1, −1/2, 1/4, −1/8, ... to 9 terms;
Find the sum of the following geometric series:
\[\frac{2}{9} - \frac{1}{3} + \frac{1}{2} - \frac{3}{4} + . . . \text { to 5 terms };\]
Find the sum of the following serie:
5 + 55 + 555 + ... to n terms;
Find the sum of the following series:
7 + 77 + 777 + ... to n terms;
Find the sum of the following series:
9 + 99 + 999 + ... to n terms;
Prove that: (91/3 . 91/9 . 91/27 ... ∞) = 3.
Prove that: (21/4 . 41/8 . 81/16. 161/32 ... ∞) = 2.
Find the rational number whose decimal expansion is `0.4bar23`.
If a, b, c are in G.P., prove that \[\frac{1}{\log_a m}, \frac{1}{\log_b m}, \frac{1}{\log_c m}\] are in A.P.
If the 4th, 10th and 16th terms of a G.P. are x, y and z respectively. Prove that x, y, z are in G.P.
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.
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.
If p, q be two A.M.'s and G be one G.M. between two numbers, then G2 =
The numbers 3, x, and x + 6 form are in G.P. Find 20th term.
For the following G.P.s, find Sn.
`sqrt(5)`, −5, `5sqrt(5)`, −25, ...
Find the sum to n terms of the sequence.
0.2, 0.02, 0.002, ...
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]
Determine whether the sum to infinity of the following G.P.s exist, if exists find them:
`1/2, 1/4, 1/8, 1/16,...`
Insert two numbers between 1 and −27 so that the resulting sequence is a G.P.
Answer the following:
Find the nth term of the sequence 0.6, 0.66, 0.666, 0.6666, ...
Answer the following:
For a G.P. if t2 = 7, t4 = 1575 find a
Find a G.P. for which sum of the first two terms is – 4 and the fifth term is 4 times the third term.
The sum of the first three terms of a G.P. is S and their product is 27. Then all such S lie in ______.
If the expansion in powers of x of the function `1/((1 - ax)(1 - bx))` is a0 + a1x + a2x2 + a3x3 ....... then an is ______.
