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)
\[\left( \frac{1}{b^2 + c^2} \right)^2 = \left( \frac{1}{b^2} \right)^2 + \frac{2}{b^2 c^2} + \left( \frac{1}{c^2} \right)^2 \]
\[ \Rightarrow \left( \frac{1}{b^2 + c^2} \right)^2 = \left( \frac{1}{ac} \right)^2 + \frac{1}{b^2 c^2} + \frac{1}{b^2 c^2} + \left( \frac{1}{bd} \right)^2 \left[ \text { Using } (1) \right]\]
\[ \Rightarrow \left( \frac{1}{b^2 + c^2} \right)^2 = \frac{1}{a^2 c^2} + \frac{1}{a^2 d^2} + \frac{1}{b^2 c^2} + \frac{1}{b^2 d^2} \left[ \text { Using }(1) \right]\]
\[ \Rightarrow \left( \frac{1}{b^2 + c^2} \right)^2 = \frac{1}{a^2}\left( \frac{1}{c^2} + \frac{1}{d^2} \right) + \frac{1}{b^2}\left( \frac{1}{c^2} + \frac{1}{d^2} \right)\]
\[ \Rightarrow \left( \frac{1}{b^2 + c^2} \right)^2 = \left( \frac{1}{a^2 + b^2} \right)\left( \frac{1}{c^2} + \frac{1}{d^2} \right)\]
\[\text{ Therefore }, \left( \frac{1}{b^2 + c^2} \right), \left( \frac{1}{c^2 + d^2} \right)\text { and } \left( \frac{1}{b^2 + c^2} \right) \text { are also in G . P } .\]
संबंधित प्रश्न
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.
If a, b, c, d are in G.P, prove that (an + bn), (bn + cn), (cn + dn) are in G.P.
Show that the sequence <an>, defined by an = \[\frac{2}{3^n}\], n ϵ N is a G.P.
Which term of the G.P.: `sqrt3, 3, 3sqrt3`, ... is 729?
Which term of the progression 18, −12, 8, ... is \[\frac{512}{729}\] ?
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 series:
\[\frac{a}{1 + i} + \frac{a}{(1 + i )^2} + \frac{a}{(1 + i )^3} + . . . + \frac{a}{(1 + i )^n} .\]
Evaluate the following:
\[\sum^n_{k = 1} ( 2^k + 3^{k - 1} )\]
Find the sum of the following series:
0.5 + 0.55 + 0.555 + ... to n terms.
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.
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.
Express the recurring decimal 0.125125125 ... as a rational number.
Find the rational numbers having the following decimal expansion:
\[3 . 5\overline 2\]
If a, b, c are in G.P., prove that the following is also in G.P.:
a2 + b2, ab + bc, b2 + c2
If (a − b), (b − c), (c − a) are in G.P., then prove that (a + b + c)2 = 3 (ab + bc + ca)
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.
Insert 5 geometric means between 16 and \[\frac{1}{4}\] .
If the sum of an infinite decreasing G.P. is 3 and the sum of the squares of its term is \[\frac{9}{2}\], then write its first term and common difference.
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
The value of 91/3 . 91/9 . 91/27 ... upto inf, is
The nth term of a G.P. is 128 and the sum of its n terms is 255. If its common ratio is 2, then its first term 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, …
For the G.P. if r = − 3 and t6 = 1701, find a.
Which term of the G.P. 5, 25, 125, 625, … is 510?
Find three numbers in G.P. such that their sum is 21 and sum of their squares is 189.
The numbers x − 6, 2x and x2 are in G.P. Find x
The numbers x − 6, 2x and x2 are in G.P. Find 1st term
For the following G.P.s, find Sn.
`sqrt(5)`, −5, `5sqrt(5)`, −25, ...
Find the sum to n terms of the sequence.
0.5, 0.05, 0.005, ...
If the common ratio of a G.P. is `2/3` and sum to infinity is 12. Find the first term
Find `sum_("r" = 0)^oo (-8)(-1/2)^"r"`
Answer the following:
Which 2 terms are inserted between 5 and 40 so that the resulting sequence is G.P.
Answer the following:
If p, q, r, s are in G.P., show that (p2 + q2 + r2) (q2 + r2 + s2) = (pq + qr + rs)2
At the end of each year the value of a certain machine has depreciated by 20% of its value at the beginning of that year. If its initial value was Rs 1250, find the value at the end of 5 years.
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 sum or difference of two G.P.s, is again a G.P.
