Advertisements
Advertisements
प्रश्न
If a, b, c, d are in G.P., prove that a2 – b2, b2 – c2, c2 – d2 are also in G.P.
Advertisements
उत्तर
Let r be the common ratio of the given G.P.
Then `b/a = c/b = d/c` = r
⇒ b = ar, c = br = ar2, d = cr = ar3
Now, a2 – b2 = a2 – a2r2
= a2(1 – r2)
b2 – c2 = a2r2 – a2r4
= a2r2 (1 – r2)
And c2 – d2 = a2r4 – a2r6
= a2r4(1 – r2)
Therefore, `(b^2 - c^2)/(a^2 - b^2) = (c^2 - d^2)/(b^2 - c^2)` = r2
Hence, a2 – b2, b2 – c2, c2 – d2 are in G.P.
APPEARS IN
संबंधित प्रश्न
Find four numbers forming a geometric progression in which third term is greater than the first term by 9, and the second term is greater than the 4th by 18.
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 and d are in G.P. show that (a2 + b2 + c2) (b2 + c2 + d2) = (ab + bc + cd)2 .
Let S be the sum, P the product and R the sum of reciprocals of n terms in a G.P. Prove that P2Rn = Sn
Show that one of the following progression is a G.P. Also, find the common ratio in case:1/2, 1/3, 2/9, 4/27, ...
Find :
nth term of the G.P.
\[\sqrt{3}, \frac{1}{\sqrt{3}}, \frac{1}{3\sqrt{3}}, . . .\]
Find the 4th term from the end of the G.P.
Which term of the progression 0.004, 0.02, 0.1, ... is 12.5?
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.
Find three numbers in G.P. whose sum is 38 and their product is 1728.
Find the sum of the following geometric progression:
2, 6, 18, ... to 7 terms;
Find the sum of the following geometric progression:
1, −1/2, 1/4, −1/8, ... to 9 terms;
Find the sum of the following geometric series:
`3/5 + 4/5^2 + 3/5^3 + 4/5^4 + ....` to 2n terms;
Prove that: (21/4 . 41/8 . 81/16. 161/32 ... ∞) = 2.
Find the rational numbers having the following decimal expansion:
\[3 . 5\overline 2\]
The sum of three numbers in G.P. is 56. If we subtract 1, 7, 21 from these numbers in that order, we obtain an A.P. Find the numbers.
If a, b, c are in G.P., prove that:
\[\frac{(a + b + c )^2}{a^2 + b^2 + c^2} = \frac{a + b + c}{a - b + c}\]
If a, b, c, d are in G.P., prove that:
(a2 + b2 + c2), (ab + bc + cd), (b2 + c2 + d2) are in 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.
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 logxa, ax/2 and logb x are in G.P., then write the value of x.
If A1, A2 be two AM's and G1, G2 be two GM's between a and b, then find the value of \[\frac{A_1 + A_2}{G_1 G_2}\]
If in an infinite G.P., first term is equal to 10 times the sum of all successive terms, then its common ratio is
The fractional value of 2.357 is
The value of 91/3 . 91/9 . 91/27 ... upto inf, is
If second term of a G.P. is 2 and the sum of its infinite terms is 8, then its first term is
If x = (43) (46) (46) (49) .... (43x) = (0.0625)−54, the value of x is
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
Check whether the following sequence is G.P. If so, write tn.
3, 4, 5, 6, …
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?
The numbers 3, x, and x + 6 form are in G.P. Find x
The numbers x − 6, 2x and x2 are in G.P. Find x
Express the following recurring decimal as a rational number:
`0.bar(7)`
Answer the following:
For a G.P. a = `4/3` and t7 = `243/1024`, find the value of r
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:
Which 2 terms are inserted between 5 and 40 so that the resulting sequence is G.P.
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 P2 R3 : S3 is equal to ______.
If the sum of an infinite GP a, ar, ar2, ar3, ...... . is 15 and the sum of the squares of its each term is 150, then the sum of ar2, ar4, ar6, .... is ______.
The sum of the first three terms of a G.P. is S and their product is 27. Then all such S lie in ______.
