Advertisements
Advertisements
प्रश्न
If pth, qth and rth terms of an A.P. and G.P. are both a, b and c respectively, show that \[a^{b - c} b^{c - a} c^{a - b} = 1\]
Advertisements
उत्तर
Let A be the first term and D be the common difference of the AP. Therefore,
\[a_p = A + \left( p - 1 \right)D = a . . . . . \left( 1 \right)\]
\[ a_q = A + \left( q - 1 \right)D = b . . . . . \left( 2 \right)\]
\[ a_r = A + \left( r - 1 \right)D = c . . . . . \left( 3 \right)\]
Also, suppose A' be the first term and R be the common ratio of the GP. Therefore,
\[a_p = A' R^{p - 1} = a . . . . . \left( 4 \right)\]
\[ a_q = A' R^{q - 1} = b . . . . . \left( 5 \right)\]
\[ a_r = A' R^{r - 1} = c . . . . . \left( 6 \right)\]
Now,
Subtracting (2) from (1), we get
\[A + \left( p - 1 \right)D - A - \left( q - 1 \right)D = a - b\]
\[ \Rightarrow \left( p - q \right)D = a - b . . . . . \left( 7 \right)\]
Subtracting (3) from (2), we get
\[A + \left( q - 1 \right)D - A - \left( r - 1 \right)D = b - c\]
\[ \Rightarrow \left( q - r \right)D = b - c . . . . . \left( 8 \right)\]
Subtracting (1) from (3), we get
\[A + \left( r - 1 \right)D - A - \left( p - 1 \right)D = c - a\]
\[ \Rightarrow \left( r - p \right)D = c - a . . . . . \left( 9 \right)\]
\[\therefore a^{b - c} b^{c - a} c^{a - b}\]
` = [A'R ^((p-1))]^((q-r)D) xx [A'R^((q-1))]^((r-p)D) xx [A'R^((r-1))]^((p-q)D) ` [Using (4), (5) (6), (7), (8) and (9)]
`= A'^((q-r)D) R^((p-1)(q-r)D) xx A'^((r-p)D) R^((q-1)(r-p)D) xx A'^((p-q)D) R^((r-1)(p-q)D) `
`=A'^[[(q-r)D+(r-p)D+(p-q)D]] xx R^[[(p-1)(q-r)D+(q-1)(r-p)D+(r-1)(p-q)D]]`
`=A'^[[q-r+r-p+p-q]D] xx R^[[pq -pr - q+r+qr-pq -r +p+pr -qr -p+q]D]`
`= (A')^0 xx R^0`
`=1 xx 1`
`= 1`
APPEARS IN
संबंधित प्रश्न
Find the 12th term of a G.P. whose 8th term is 192 and the common ratio is 2.
For what values of x, the numbers `-2/7, x, -7/2` are in G.P?
Find a G.P. for which sum of the first two terms is –4 and the fifth term is 4 times the third term.
Find the value of n so that `(a^(n+1) + b^(n+1))/(a^n + b^n)` may be the geometric mean between a and b.
The sum of two numbers is 6 times their geometric mean, show that numbers are in the ratio `(3 + 2sqrt2) ":" (3 - 2sqrt2)`.
The first term of a G.P. is 1. The sum of the third term and fifth term is 90. Find the common ratio of G.P.
If a and b are the roots of are roots of x2 – 3x + p = 0 , and c, d are roots of x2 – 12x + q = 0, where a, b, c, d, form a G.P. Prove that (q + p): (q – p) = 17 : 15.
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 4th term from the end of the G.P.
Which term of the G.P. :
\[2, 2\sqrt{2}, 4, . . .\text { is }128 ?\]
If a, b, c, d and p are different real numbers such that:
(a2 + b2 + c2) p2 − 2 (ab + bc + cd) p + (b2 + c2 + d2) ≤ 0, then show that a, b, c and d are in G.P.
The sum of first three terms of a G.P. is 13/12 and their product is − 1. Find the G.P.
Let an be the nth term of the G.P. of positive numbers.
Let \[\sum^{100}_{n = 1} a_{2n} = \alpha \text { and } \sum^{100}_{n = 1} a_{2n - 1} = \beta,\] such that α ≠ β. Prove that the common ratio of the G.P. is α/β.
Prove that: (91/3 . 91/9 . 91/27 ... ∞) = 3.
Express the recurring decimal 0.125125125 ... as a rational number.
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 + 2b + 2c) (a − 2b + 2c) = a2 + 4c2.
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, 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. and a, b, d are in G.P., then prove that a, a − b, d − c are in G.P.
If xa = xb/2 zb/2 = zc, then prove that \[\frac{1}{a}, \frac{1}{b}, \frac{1}{c}\] are in A.P.
Find the geometric means of the following pairs of number:
a3b and ab3
If logxa, ax/2 and logb x are in G.P., then write the value of x.
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 ______.
Check whether the following sequence is G.P. If so, write tn.
7, 14, 21, 28, …
Mosquitoes are growing at a rate of 10% a year. If there were 200 mosquitoes in the beginning. Write down the number of mosquitoes after 3 years.
For a G.P. if a = 2, r = 3, Sn = 242 find n
Find the sum to n terms of the sequence.
0.5, 0.05, 0.005, ...
Find the sum to n terms of the sequence.
0.2, 0.02, 0.002, ...
If S, P, R are the sum, product, and sum of the reciprocals of n terms of a G.P. respectively, then verify that `["S"/"R"]^"n"` = P2
If Sn, S2n, S3n are the sum of n, 2n, 3n terms of a G.P. respectively, then verify that Sn (S3n – S2n) = (S2n – Sn)2.
Express the following recurring decimal as a rational number:
`51.0bar(2)`
Find : `sum_("n" = 1)^oo 0.4^"n"`
Select the correct answer from the given alternative.
The tenth term of the geometric sequence `1/4, (-1)/2, 1, -2,` ... is –
Select the correct answer from the given alternative.
Which of the following is not true, where A, G, H are the AM, GM, HM of a and b respectively. (a, b > 0)
The sum or difference of two G.P.s, is again a G.P.
If `e^((cos^2x + cos^4x + cos^6x + ...∞)log_e2` satisfies the equation t2 – 9t + 8 = 0, then the value of `(2sinx)/(sinx + sqrt(3)cosx)(0 < x ,< π/2)` 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 ______.
