Advertisements
Advertisements
प्रश्न
If a, b, c, d are in G.P., prove that:
\[\frac{1}{a^2 + b^2}, \frac{1}{b^2 - c^2}, \frac{1}{c^2 + d^2} \text { are in G . P } .\]
Advertisements
उत्तर
a, b, c and d are in G.P.
\[\therefore b^2 = ac\]
\[ad = bc \]
\[ c^2 = bd\] .......(1)
\[\left( \frac{1}{b^2 + c^2} \right)^2 = \left( \frac{1}{b^2} \right)^2 + \frac{2}{b^2 c^2} + \left( \frac{1}{c^2} \right)^2 \]
\[ \Rightarrow \left( \frac{1}{b^2 + c^2} \right)^2 = \left( \frac{1}{ac} \right)^2 + \frac{1}{b^2 c^2} + \frac{1}{b^2 c^2} + \left( \frac{1}{bd} \right)^2 \left[ \text { Using } (1) \right]\]
\[ \Rightarrow \left( \frac{1}{b^2 + c^2} \right)^2 = \frac{1}{a^2 c^2} + \frac{1}{a^2 d^2} + \frac{1}{b^2 c^2} + \frac{1}{b^2 d^2} \left[ \text { Using }(1) \right]\]
\[ \Rightarrow \left( \frac{1}{b^2 + c^2} \right)^2 = \frac{1}{a^2}\left( \frac{1}{c^2} + \frac{1}{d^2} \right) + \frac{1}{b^2}\left( \frac{1}{c^2} + \frac{1}{d^2} \right)\]
\[ \Rightarrow \left( \frac{1}{b^2 + c^2} \right)^2 = \left( \frac{1}{a^2 + b^2} \right)\left( \frac{1}{c^2} + \frac{1}{d^2} \right)\]
\[\text{ Therefore }, \left( \frac{1}{b^2 + c^2} \right), \left( \frac{1}{c^2 + d^2} \right)\text { and } \left( \frac{1}{b^2 + c^2} \right) \text { are also in G . P } .\]
APPEARS IN
संबंधित प्रश्न
The 5th, 8th and 11th terms of a G.P. are p, q and s, respectively. Show that q2 = ps.
Which term of the following sequence:
`2, 2sqrt2, 4,.... is 128`
Find the sum of the products of the corresponding terms of the sequences `2, 4, 8, 16, 32 and 128, 32, 8, 2, 1/2`
Show that the sequence <an>, defined by an = \[\frac{2}{3^n}\], n ϵ N is a G.P.
Find:
the 10th term of the G.P.
\[- \frac{3}{4}, \frac{1}{2}, - \frac{1}{3}, \frac{2}{9}, . . .\]
Find :
the 8th term of the G.P. 0.3, 0.06, 0.012, ...
Find :
nth term of the G.P.
\[\sqrt{3}, \frac{1}{\sqrt{3}}, \frac{1}{3\sqrt{3}}, . . .\]
Which term of the G.P. :
\[2, 2\sqrt{2}, 4, . . .\text { is }128 ?\]
If the G.P.'s 5, 10, 20, ... and 1280, 640, 320, ... have their nth terms equal, find the value of n.
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.
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.
Find the sum of the following series to infinity:
10 − 9 + 8.1 − 7.29 + ... ∞
Prove that: (91/3 . 91/9 . 91/27 ... ∞) = 3.
If Sp denotes the sum of the series 1 + rp + r2p + ... to ∞ and sp the sum of the series 1 − rp + r2p − ... to ∞, prove that Sp + sp = 2 . S2p.
If a, b, c are in G.P., prove that log a, log b, log c are in A.P.
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 (b2 + c2) = c (a2 + b2)
If a, b, c are in G.P., prove that the following is also in G.P.:
a3, b3, c3
If a, b, c are in A.P. and a, b, d are in G.P., then prove that a, a − b, d − c are in G.P.
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 \[\frac{1}{a + b}, \frac{1}{2b}, \frac{1}{b + c}\] 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., b,c,d are in G.P. and \[\frac{1}{c}, \frac{1}{d}, \frac{1}{e}\] are in A.P., prove that a, c,e are in G.P.
Find the geometric means of the following pairs of number:
−8 and −2
If S be the sum, P the product and R be the sum of the reciprocals of n terms of a GP, then P2 is equal to
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 \[\frac{1}{1 + x} - \frac{1 - x}{(1 + x )^2} + \frac{(1 - x )^2}{(1 + x )^3} - \frac{(1 - x )^3}{(1 + x )^4} + . . . . . . is\]
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
The numbers 3, x, and x + 6 form are in G.P. Find 20th term.
The numbers x − 6, 2x and x2 are in G.P. Find nth term
For a G.P. sum of first 3 terms is 125 and sum of next 3 terms is 27, find the value of r
Find: `sum_("r" = 1)^10(3 xx 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
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`
Answer the following:
Find five numbers in G.P. such that their product is 243 and sum of second and fourth number is 10.
Answer the following:
Find `sum_("r" = 1)^"n" (2/3)^"r"`
Answer the following:
If a, b, c are in G.P. and ax2 + 2bx + c = 0 and px2 + 2qx + r = 0 have common roots then verify that pb2 – 2qba + ra2 = 0
The sum or difference of two G.P.s, is again a G.P.
The sum of the first three terms of a G.P. is S and their product is 27. Then all such S lie in ______.
