Advertisements
Advertisements
प्रश्न
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.
Advertisements
उत्तर
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
संबंधित प्रश्न
Find the sum to indicated number of terms of the geometric progressions `sqrt7, sqrt21,3sqrt7`...n terms.
How many terms of G.P. 3, 32, 33, … are needed to give the sum 120?
The sum of first three terms of a G.P. is 16 and the sum of the next three terms is 128. Determine the first term, the common ratio and the sum to n terms of the G.P.
if `(a+ bx)/(a - bx) = (b +cx)/(b - cx) = (c + dx)/(c- dx) (x != 0)` then show that a, b, c and d are in G.P.
Find :
the 8th term of the G.P. 0.3, 0.06, 0.012, ...
Find :
the 10th term of the G.P.
\[\sqrt{2}, \frac{1}{\sqrt{2}}, \frac{1}{2\sqrt{2}}, . . .\]
Which term of the G.P.: `sqrt3, 3, 3sqrt3`, ... is 729?
If 5th, 8th and 11th terms of a G.P. are p. q and s respectively, prove that q2 = ps.
The sum of first three terms of a G.P. is \[\frac{39}{10}\] and their product is 1. Find the common ratio and the terms.
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.
Find the sum of the following geometric progression:
1, 3, 9, 27, ... to 8 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 series:
0.6 + 0.66 + 0.666 + .... to n terms
The common ratio of a G.P. is 3 and the last term is 486. If the sum of these terms be 728, find the first term.
If S1, S2, S3 be respectively the sums of n, 2n, 3n terms of a G.P., then prove that \[S_1^2 + S_2^2\] = S1 (S2 + S3).
How many terms of the G.P. `3, 3/2, 3/4` ..... are needed to give the sum `3069/512`?
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.
Prove that: (21/4 . 41/8 . 81/16. 161/32 ... ∞) = 2.
One side of an equilateral triangle is 18 cm. The mid-points of its sides are joined to form another triangle whose mid-points, in turn, are joined to form still another triangle. The process is continued indefinitely. Find the sum of the (i) perimeters of all the triangles. (ii) areas of all triangles.
If a, b, c are three distinct real numbers in G.P. and a + b + c = xb, then prove that either x< −1 or x > 3.
If a = 1 + b + b2 + b3 + ... to ∞, then write b in terms of a.
The fractional value of 2.357 is
The product (32), (32)1/6 (32)1/36 ... to ∞ is equal to
Check whether the following sequence is G.P. If so, write tn.
1, –5, 25, –125 …
The fifth term of a G.P. is x, eighth term of a G.P. is y and eleventh term of a G.P. is z verify whether y2 = xz
The numbers 3, x, and x + 6 form are in G.P. Find nth term
For the following G.P.s, find Sn.
`sqrt(5)`, −5, `5sqrt(5)`, −25, ...
For a G.P. if S5 = 1023 , r = 4, Find a
If Sn, S2n, S3n are the sum of n, 2n, 3n terms of a G.P. respectively, then verify that Sn (S3n – S2n) = (S2n – Sn)2.
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 perimeters of all the squares
Find GM of two positive numbers whose A.M. and H.M. are 75 and 48
Select the correct answer from the given alternative.
The common ratio for the G.P. 0.12, 0.24, 0.48, is –
Answer the following:
If pth, qth and rth terms of a G.P. are x, y, z respectively. Find the value of xq–r .yr–p .zp–q
In a G.P. of positive terms, if any term is equal to the sum of the next two terms. Then the common ratio of the G.P. is ______.
The third term of G.P. is 4. The product of its first 5 terms is ______.
The third term of a G.P. is 4, the product of the first five terms is ______.
The sum of infinite number of terms of a decreasing G.P. is 4 and the sum of the terms to m squares of its terms to infinity is `16/3`, then the G.P. is ______.
Let A1, A2, A3, .... be an increasing geometric progression of positive real numbers. If A1A3A5A7 = `1/1296` and A2 + A4 = `7/36`, then the value of A6 + A8 + A10 is equal to ______.
