Advertisements
Advertisements
Question
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
Solution
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`
RELATED QUESTIONS
Which term of the following sequence:
`2, 2sqrt2, 4,.... is 128`
Find the sum to indicated number of terms of the geometric progressions `sqrt7, sqrt21,3sqrt7`...n terms.
Find a G.P. for which sum of the first two terms is –4 and the fifth term is 4 times the third term.
Show that the products of the corresponding terms of the sequences a, ar, ar2, …arn – 1 and A, AR, AR2, … `AR^(n-1)` form a G.P, and find the common ratio
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.
Find :
nth term of the G.P.
\[\sqrt{3}, \frac{1}{\sqrt{3}}, \frac{1}{3\sqrt{3}}, . . .\]
The fourth term of a G.P. is 27 and the 7th term is 729, find the G.P.
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.
Evaluate the following:
\[\sum^n_{k = 1} ( 2^k + 3^{k - 1} )\]
Evaluate the following:
\[\sum^{10}_{n = 2} 4^n\]
Find the sum of the following series:
7 + 77 + 777 + ... to n terms;
Find the sum of the following series:
9 + 99 + 999 + ... to n terms;
Find the sum of the following series:
0.6 + 0.66 + 0.666 + .... to n terms
How many terms of the G.P. 3, 3/2, 3/4, ... be taken together to make \[\frac{3069}{512}\] ?
A person has 2 parents, 4 grandparents, 8 great grandparents, and so on. Find the number of his ancestors during the ten generations preceding his own.
Find the sum of the following serie to infinity:
8 + \[4\sqrt{2}\] + 4 + ... ∞
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
\[\frac{2S S_1}{S^2 + S_1}\text { and } \frac{S^2 - S_1}{S^2 + S_1}\]
The sum of three numbers which are consecutive terms of an A.P. is 21. If the second number is reduced by 1 and the third is increased by 1, we obtain three consecutive terms of a G.P. Find the numbers.
If a, b, c are in G.P., prove that:
\[\frac{1}{a^2 - b^2} + \frac{1}{b^2} = \frac{1}{b^2 - c^2}\]
If (a − b), (b − c), (c − a) are in G.P., then prove that (a + b + c)2 = 3 (ab + bc + ca)
Find the geometric means of the following pairs of number:
a3b and ab3
If pth, qth and rth terms of an A.P. are in G.P., then the common ratio of this G.P. is
The sum of an infinite G.P. is 4 and the sum of the cubes of its terms is 92. The common ratio of the original G.P. is
Check whether the following sequence is G.P. If so, write tn.
3, 4, 5, 6, …
Determine whether the sum to infinity of the following G.P.s exist, if exists find them:
`2, 4/3, 8/9, 16/27, ...`
The sum of an infinite G.P. is 5 and the sum of the squares of these terms is 15 find the G.P.
Find : `sum_("r" = 1)^oo 4(0.5)^"r"`
Find `sum_("r" = 0)^oo (-8)(-1/2)^"r"`
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 common ratio of the G.P is 5, 5th term is 1875, the first term is -
Answer the following:
For a G.P. a = `4/3` and t7 = `243/1024`, find the value of r
Answer the following:
Find three numbers in G.P. such that their sum is 35 and their product is 1000
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:
If p, q, r, s are in G.P., show that (pn + qn), (qn + rn) , (rn + sn) are also in G.P.
If the pth and qth terms of a G.P. are q and p respectively, show that its (p + q)th term is `(q^p/p^q)^(1/(p - q))`
Let `{a_n}_(n = 0)^∞` be a sequence such that a0 = a1 = 0 and an+2 = 2an+1 – an + 1 for all n ≥ 0. Then, `sum_(n = 2)^∞ a^n/7^n` is equal to ______.
If the expansion in powers of x of the function `1/((1 - ax)(1 - bx))` is a0 + a1x + a2x2 + a3x3 ....... then an is ______.
