Advertisements
Advertisements
प्रश्न
If a, b, c, d are in G.P., prove that:
(a2 + b2), (b2 + c2), (c2 + d2) 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)
\[\left( b^2 + c^2 \right)^2 = \left( b^2 \right)^2 + 2 b^2 c^2 + \left( c^2 \right)^2 \]
\[ \Rightarrow \left( b^2 + c^2 \right)^2 = \left( ac \right)^2 + b^2 c^2 + b^2 c^2 + \left( bd \right)^2 \left[\text { Using } (1) \right]\]
\[ \Rightarrow \left( b^2 + c^2 \right)^2 = a^2 c^2 + a^2 d^2 + b^2 c^2 + b^2 d^2 \left[ \text { Using } (1) \right]\]
\[ \Rightarrow \left( b^2 + c^2 \right)^2 = a^2 \left( c^2 + d^2 \right) + b^2 \left( c^2 + d^2 \right)\]
\[ \Rightarrow \left( b^2 + c^2 \right)^2 = \left( a^2 + b^2 \right)\left( c^2 + d^2 \right)\]
\[\text {Therefore, } \left( a^2 + b^2 \right), \left( c^2 + d^2 \right)\text{ and } \left( b^2 + c^2 \right) \text { are also in G . P } .\]
APPEARS IN
संबंधित प्रश्न
Which term of the following sequence:
`2, 2sqrt2, 4,.... is 128`
Find the sum to indicated number of terms in the geometric progressions 1, – a, a2, – a3, ... n terms (if a ≠ – 1).
Find the sum to indicated number of terms in the geometric progressions x3, x5, x7, ... n terms (if x ≠ ± 1).
Find the sum to n terms of the sequence, 8, 88, 888, 8888… .
Show that one of the following progression is a G.P. Also, find the common ratio in case:
4, −2, 1, −1/2, ...
Show that one of the following progression is a G.P. Also, find the common ratio in case:
−2/3, −6, −54, ...
Find :
the 10th term of the G.P.
\[\sqrt{2}, \frac{1}{\sqrt{2}}, \frac{1}{2\sqrt{2}}, . . .\]
Find the 4th term from the end of the G.P.
\[\frac{1}{2}, \frac{1}{6}, \frac{1}{18}, \frac{1}{54}, . . . , \frac{1}{4374}\]
Find the sum of the following geometric series:
\[\frac{a}{1 + i} + \frac{a}{(1 + i )^2} + \frac{a}{(1 + i )^3} + . . . + \frac{a}{(1 + i )^n} .\]
Find the sum of the following series:
9 + 99 + 999 + ... to n terms;
If a and b are the roots of x2 − 3x + p = 0 and c, d are the roots x2 − 12x + q = 0, where a, b, c, d form a G.P. Prove that (q + p) : (q − p) = 17 : 15.
How many terms of the G.P. `3, 3/2, 3/4` ..... are needed to give the sum `3069/512`?
Find the sum of 2n terms of the series whose every even term is 'a' times the term before it and every odd term is 'c' times the term before it, the first term being unity.
Find the sum of the following serie to infinity:
8 + \[4\sqrt{2}\] + 4 + ... ∞
Find the sum of the following series to infinity:
10 − 9 + 8.1 − 7.29 + ... ∞
Prove that: (21/4 . 41/8 . 81/16. 161/32 ... ∞) = 2.
Find the rational numbers having the following decimal expansion:
\[0 .\overline {231 }\]
Find the rational numbers having the following decimal expansion:
\[3 . 5\overline 2\]
The sum of first two terms of an infinite G.P. is 5 and each term is three times the sum of the succeeding terms. Find the G.P.
Find k such that k + 9, k − 6 and 4 form three consecutive terms of a G.P.
If a, b, c, d are in G.P., prove that:
\[\frac{ab - cd}{b^2 - c^2} = \frac{a + c}{b}\]
If a, b, c are in G.P., then prove that:
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 pth, qth and rth terms of an A.P. and G.P. are both a, b and c respectively, show that \[a^{b - c} b^{c - a} c^{a - b} = 1\]
If (p + q)th and (p − q)th terms of a G.P. are m and n respectively, then write is pth term.
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 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 \[\frac{1}{1 + x} - \frac{1 - x}{(1 + x )^2} + \frac{(1 - x )^2}{(1 + x )^3} - \frac{(1 - x )^3}{(1 + x )^4} + . . . . . . is\]
The number of bacteria in a culture doubles every hour. If there were 50 bacteria originally in the culture, how many bacteria will be there at the end of 5th hour?
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?
The numbers x − 6, 2x and x2 are in G.P. Find nth term
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]
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
Answer the following:
For a G.P. a = `4/3` and t7 = `243/1024`, find the value of r
Answer the following:
For a sequence Sn = 4(7n – 1) verify that the sequence is a G.P.
Answer the following:
Find k so that k – 1, k, k + 2 are consecutive terms of a G.P.
The third term of a G.P. is 4, the product of the first five terms is ______.
For an increasing G.P. a1, a2 , a3 ........., an, if a6 = 4a4, a9 – a7 = 192, then the value of `sum_(i = 1)^∞ 1/a_i` is ______.
