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 sum to indicated number of terms of the geometric progressions `sqrt7, sqrt21,3sqrt7`...n terms.
Evaluate `sum_(k=1)^11 (2+3^k )`
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`.
Show that one of the following progression is a G.P. Also, find the common ratio in case:
\[a, \frac{3 a^2}{4}, \frac{9 a^3}{16}, . . .\]
Find :
the 12th term of the G.P.
\[\frac{1}{a^3 x^3}, ax, a^5 x^5 , . . .\]
Find the 4th term from the end of the G.P.
If the G.P.'s 5, 10, 20, ... and 1280, 640, 320, ... have their nth terms equal, find the value of n.
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.
If the pth and qth terms of a G.P. are q and p, respectively, then show that (p + q)th term is \[\left( \frac{q^p}{p^q} \right)^\frac{1}{p - q}\].
The product of three numbers in G.P. is 125 and the sum of their products taken in pairs is \[87\frac{1}{2}\] . Find them.
The sum of three numbers in G.P. is 14. If the first two terms are each increased by 1 and the third term decreased by 1, the resulting numbers are in A.P. Find the numbers.
The product of three numbers in G.P. is 216. If 2, 8, 6 be added to them, the results are in A.P. Find the numbers.
Find the sum of the following geometric progression:
(a2 − b2), (a − b), \[\left( \frac{a - b}{a + b} \right)\] to n terms;
Find the sum of the following geometric series:
0.15 + 0.015 + 0.0015 + ... to 8 terms;
Show that the ratio of the sum of first n terms of a G.P. to the sum of terms from (n + 1)th to (2n)th term is \[\frac{1}{r^n}\].
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`?
One side of an equilateral triangle is 18 cm. The mid-points of its sides are joined to form another triangle whose mid-points, in turn, are joined to form still another triangle. The process is continued indefinitely. Find the sum of the (i) perimeters of all the triangles. (ii) areas of all triangles.
Find an infinite G.P. whose first term is 1 and each term is the sum of all the terms which follow it.
If S denotes the sum of an infinite G.P. S1 denotes the sum of the squares of its terms, then prove that the first term and common ratio are respectively
\[\frac{2S S_1}{S^2 + S_1}\text { and } \frac{S^2 - S_1}{S^2 + S_1}\]
If a, b, c are in G.P., prove that:
(a + 2b + 2c) (a − 2b + 2c) = a2 + 4c2.
If \[\frac{1}{a + b}, \frac{1}{2b}, \frac{1}{b + c}\] are three consecutive terms of an A.P., prove that a, b, c are the three consecutive terms of a G.P.
Find the geometric means of the following pairs of number:
2 and 8
Find the geometric means of the following pairs of number:
−8 and −2
If the fifth term of a G.P. is 2, then write the product of its 9 terms.
If logxa, ax/2 and logb x are in G.P., then write the value of x.
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 a = 1 + b + b2 + b3 + ... to ∞, then write b in terms of a.
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
If pth, qth and rth terms of an A.P. are in G.P., then the common ratio of this G.P. is
If x is positive, the sum to infinity of the series \[\frac{1}{1 + x} - \frac{1 - x}{(1 + x )^2} + \frac{(1 - x )^2}{(1 + x )^3} - \frac{(1 - x )^3}{(1 + x )^4} + . . . . . . is\]
Check whether the following sequence is G.P. If so, write tn.
`sqrt(5), 1/sqrt(5), 1/(5sqrt(5)), 1/(25sqrt(5))`, ...
For the G.P. if a = `7/243`, r = 3 find t6.
The numbers 3, x, and x + 6 form are in G.P. Find 20th term.
If the common ratio of a G.P. is `2/3` and sum to infinity is 12. Find the first term
Find GM of two positive numbers whose A.M. and H.M. are 75 and 48
Answer the following:
In a G.P., the fourth term is 48 and the eighth term is 768. Find the tenth term
Find a G.P. for which sum of the first two terms is – 4 and the fifth term is 4 times the third term.
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 ______.
