Advertisements
Advertisements
Question
Find the value of n so that `(a^(n+1) + b^(n+1))/(a^n + b^n)` may be the geometric mean between a and b.
Solution
The geometric mean between a and b = `sqrt"ab"`
⇒ `("a"^("n"+ 1) + "b"^("n" + 1))/("a"^"n" + "b"^"n") = sqrt"ab"`
∴ `"a"^("n"+ 1) + "b"^("n" + 1) = sqrt"ab" ("a"^"n" + "b"^"n")`
= `"a"^("n"+ 1/2) "b"^(1/2) + "a"^(1/2) "b"^("n" + 1/2)`
or `("a"^("n" + 1) - "a"^("n" + 1/2) "b"^(1/2)) - ("a"^(1/2) "b"^("n" + 1/2) - "b"^("n" + 1)) = 0`
or `"a"^("n" + 1/2) ("a"^(1/2) - "b"^(1/2)) - "b"^ ("n" + 1/2)("a" ^(1/2) - "b"^(1/2)) = 0`
or `("a"^(1/2) - "b"^(1/2)) ("a"^("n" + 1/2) - "b"^ ("n" + 1/2)) = 0`
`"a" ^(1/2) - "b"^(1/2) ≠ 0`
∴ `"a"^("n" + 1/2) - "b"^ ("n" + 1/2) = 0`
or `"a"^("n" + 1/2) = "b"^ ("n" + 1/2)`
or `("a"/"b")^("n"+1/2) = 1 = ("a"/"b")^0`
⇒ `"n"+ 1/2 = 0`
n = `(-1)/2`
APPEARS IN
RELATED QUESTIONS
Find the sum to indicated number of terms in the geometric progressions x3, x5, x7, ... n terms (if x ≠ ± 1).
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.
The first term of a G.P. is 1. The sum of the third term and fifth term is 90. Find the common ratio of G.P.
Find :
the 8th term of the G.P. 0.3, 0.06, 0.012, ...
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 three numbers in G.P. whose product is 729 and the sum of their products in pairs is 819.
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:
x3, x5, x7, ... to n terms
Evaluate the following:
\[\sum^{10}_{n = 2} 4^n\]
How many terms of the series 2 + 6 + 18 + ... must be taken to make the sum equal to 728?
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 following serie to infinity:
`2/5 + 3/5^2 +2/5^3 + 3/5^4 + ... ∞.`
Prove that: (91/3 . 91/9 . 91/27 ... ∞) = 3.
Find the rational numbers having the following decimal expansion:
\[0 . \overline3\]
Three numbers are in A.P. and their sum is 15. If 1, 3, 9 be added to them respectively, they form a G.P. Find the numbers.
If a, b, c are in G.P., prove that:
\[a^2 b^2 c^2 \left( \frac{1}{a^3} + \frac{1}{b^3} + \frac{1}{c^3} \right) = a^3 + b^3 + c^3\]
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 \[\frac{1}{a + b}, \frac{1}{2b}, \frac{1}{b + c}\] are three consecutive terms of an A.P., prove that a, b, c are the three consecutive terms of a G.P.
If 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.
Find the geometric means of the following pairs of number:
2 and 8
If A1, A2 be two AM's and G1, G2 be two GM's between a and b, then find the value of \[\frac{A_1 + A_2}{G_1 G_2}\]
The nth term of a G.P. is 128 and the sum of its n terms is 225. If its common ratio is 2, then its first term is
If p, q be two A.M.'s and G be one G.M. between two numbers, then G2 =
If x is positive, the sum to infinity of the series \[\frac{1}{1 + x} - \frac{1 - x}{(1 + x )^2} + \frac{(1 - x )^2}{(1 + x )^3} - \frac{(1 - x )^3}{(1 + x )^4} + . . . . . . is\]
In a G.P. if the (m + n)th term is p and (m − n)th term is q, then its mth term is
For the G.P. if a = `7/243`, r = 3 find t6.
Which term of the G.P. 5, 25, 125, 625, … is 510?
If for a sequence, tn = `(5^("n"-3))/(2^("n"-3))`, show that the sequence is a G.P. Find its first term and the common ratio
A ball is dropped from a height of 80 ft. The ball is such that it rebounds `(3/4)^"th"` of the height it has fallen. How high does the ball rebound on 6th bounce? How high does the ball rebound on nth bounce?
Express the following recurring decimal as a rational number:
`0.bar(7)`
Find `sum_("r" = 0)^oo (-8)(-1/2)^"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 areas of all the squares
Select the correct answer from the given alternative.
The common ratio for the G.P. 0.12, 0.24, 0.48, is –
Select the correct answer from the given alternative.
If for a G.P. `"t"_6/"t"_3 = 1458/54` then 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))`
The third term of a G.P. is 4, the product of the first five terms is ______.
If the expansion in powers of x of the function `1/((1 - ax)(1 - bx))` is a0 + a1x + a2x2 + a3x3 ....... then an is ______.
