Advertisements
Advertisements
प्रश्न
If a, b, c are in G.P., prove that:
(a + 2b + 2c) (a − 2b + 2c) = a2 + 4c2.
Advertisements
उत्तर
a, b and c are in G.P.
\[\therefore b^2 = ac\] .......(1)
\[\text { LHS }= \left( a + 2b + 2c \right)\left( a - 2b + 2c \right)\]
\[ = a^2 - 4 b^2 + 4 c^2 + 4ac\]
\[ = a^2 - 4ac + 4 c^2 + 4ac \left[ \text { Using }(1) \right]\]
\[ = a^2 + 4 c^2 = \text { RHS }\]
APPEARS IN
संबंधित प्रश्न
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`
If the pth, qth and rth terms of a G.P. are a, b and c, respectively. Prove that `a^(q - r) b^(r-p) c^(p-q) = 1`.
If a, b, c, d are in G.P, prove that (an + bn), (bn + cn), (cn + dn) are in G.P.
Which term of the G.P. :
\[\frac{1}{3}, \frac{1}{9}, \frac{1}{27} . . \text { . is } \frac{1}{19683} ?\]
The fourth term of a G.P. is 27 and the 7th term is 729, find the G.P.
The seventh term of a G.P. is 8 times the fourth term and 5th term is 48. Find the G.P.
In a GP the 3rd term is 24 and the 6th term is 192. Find the 10th term.
Find three numbers in G.P. whose sum is 38 and their product is 1728.
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.
The 4th and 7th terms of a G.P. are \[\frac{1}{27} \text { and } \frac{1}{729}\] respectively. Find the sum of n terms of the G.P.
Find the sum of the following serie to infinity:
8 + \[4\sqrt{2}\] + 4 + ... ∞
If (a − b), (b − c), (c − a) are in G.P., then prove that (a + b + c)2 = 3 (ab + bc + ca)
If xa = xb/2 zb/2 = zc, then prove that \[\frac{1}{a}, \frac{1}{b}, \frac{1}{c}\] are in A.P.
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.
The fractional value of 2.357 is
If the sum of first two terms of an infinite GP is 1 every term is twice the sum of all the successive terms, then its first term is
If a, b, c are in G.P. and x, y are AM's between a, b and b,c respectively, then
In a G.P. of even number of terms, the sum of all terms is five times the sum of the odd terms. The common ratio of the G.P. is
Mark the correct alternative in the following question:
Let S be the sum, P be the product and R be the sum of the reciprocals of 3 terms of a G.P. Then p2R3 : S3 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 a G.P. If t4 = 16, t9 = 512, find S10
Find the sum to n terms of the sequence.
0.2, 0.02, 0.002, ...
For a sequence, if Sn = 2(3n –1), find the nth term, hence show that the sequence is a G.P.
Find: `sum_("r" = 1)^10(3 xx 2^"r")`
Determine whether the sum to infinity of the following G.P.s exist, if exists find them:
`2, 4/3, 8/9, 16/27, ...`
Determine whether the sum to infinity of the following G.P.s exist, if exists find them:
9, 8.1, 7.29, ...
Express the following recurring decimal as a rational number:
`2.3bar(5)`
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
If the A.M. of two numbers exceeds their G.M. by 2 and their H.M. by `18/5`, find the numbers.
Select the correct answer from the given alternative.
If for a G.P. `"t"_6/"t"_3 = 1458/54` then r = ?
Select the correct answer from the given alternative.
Which term of the geometric progression 1, 2, 4, 8, ... is 2048
Answer the following:
For a sequence , if tn = `(5^("n" - 2))/(7^("n" - 3))`, verify whether the sequence is a G.P. If it is a G.P., find its first term and the common ratio.
Answer the following:
Find three numbers in G.P. such that their sum is 35 and their product is 1000
Answer the following:
For a G.P. if t2 = 7, t4 = 1575 find a
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 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 ______.
If 0 < x, y, a, b < 1, then the sum of the infinite terms of the series `sqrt(x)(sqrt(a) + sqrt(x)) + sqrt(x)(sqrt(ab) + sqrt(xy)) + sqrt(x)(bsqrt(a) + ysqrt(x)) + ...` is ______.
