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 ab–c . bc – a . ca – b = 1
Advertisements
Solution
Let A and d be the first term and common difference respectively of an A.P. and x and R be the first term and common ratio respectively of the G.P.
∴ A + (p – 1)d = a .....(i)
A + (q – 1)d = b .....(ii)
And A + (r – 1)d = c ......(iii)
For G.P., we have
xRp–1 = a .....(iv)
xRq–1 = b .....(v)
And xRr–1 = c .....(vi)
Subtracting equation (ii) from equation (i) we get
(p – q)d = a – b ......(vii)
Similarly, (q – r)d = b – c ......(viii)
And (r – p)d = c – a ......(ix)
Now we have to prove that
ab–c . bc–a . ca–b = 1
L.H.S. ab–c . bc–a . ca–b
= `[x"R"^(p - 1)]^((q - r)d) * [x"R"^(q - 1)]^((r - p)d) * [x"R"^(r - 1)]^((p - q)d)` ....[From (i), (ii), (iii), (iv), (v), (vi), (vii), (viii), (ix)]
= `x^((q - r)d) * "R"^((p - 1) (q - r)d) * x^((r - p)d) * "R"^((q - 1) (r - p)d) * x^((p - q)d) * "R"^((r - 1)(p - q)d)`
= `x^((q - r)d + (r - p)d) "R"^((p - 1)(q - r)d + (q - 1)(r - p)d + (r - 1)(p - q)d)`
= `x^((q-r + r - p + p - q)d) * "R"^((pq - pr - q + r + qr - pq - r + p + pr + pr - qr - p + q)d)`
= `x^((0)d) * "R"^((0)d)`
= `x^0 * "R"^0`
= 1 R.H.S.
L.H.S. = R.H.S.
Hence proved.
APPEARS IN
RELATED QUESTIONS
Find the 12th term of a G.P. whose 8th term is 192 and the common ratio is 2.
The 5th, 8th and 11th terms of a G.P. are p, q and s, respectively. Show that q2 = ps.
Find the sum to indicated number of terms in the geometric progressions 1, – a, a2, – a3, ... n terms (if a ≠ – 1).
The sum of first three terms of a G.P. is 16 and the sum of the next three terms is 128. Determine the first term, the common ratio and the sum to n terms of the G.P.
Given a G.P. with a = 729 and 7th term 64, determine S7.
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 `1/r^n`.
Show that one of the following progression is a G.P. Also, find the common ratio in case:
\[a, \frac{3 a^2}{4}, \frac{9 a^3}{16}, . . .\]
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 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.
Find the sum of the following geometric progression:
4, 2, 1, 1/2 ... to 10 terms.
Find the sum of the following geometric series:
(x +y) + (x2 + xy + y2) + (x3 + x2y + xy2 + y3) + ... to n terms;
Find the sum of the following geometric series:
`sqrt7, sqrt21, 3sqrt7,...` to n terms
Evaluate the following:
\[\sum^n_{k = 1} ( 2^k + 3^{k - 1} )\]
Find the sum of the following series:
0.5 + 0.55 + 0.555 + ... to n terms.
How many terms of the G.P. 3, \[\frac{3}{2}, \frac{3}{4}\] ..... are needed to give the sum \[\frac{3069}{512}\] ?
The sum of first two terms of an infinite G.P. is 5 and each term is three times the sum of the succeeding terms. Find the G.P.
Show that in an infinite G.P. with common ratio r (|r| < 1), each term bears a constant ratio to the sum of all terms that follow it.
If a, b, c are in G.P., prove that log a, log b, log c are in A.P.
If a, b, c are in G.P., then prove that:
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
The product (32), (32)1/6 (32)1/36 ... to ∞ is equal to
The two geometric means between the numbers 1 and 64 are
If for a sequence, tn = `(5^("n"-3))/(2^("n"-3))`, show that the sequence is a G.P. Find its first term and the common ratio
The numbers 3, x, and x + 6 form are in G.P. Find 20th term.
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 n years.
The numbers x − 6, 2x and x2 are in G.P. Find x
For a G.P. if a = 2, r = 3, Sn = 242 find n
Determine whether the sum to infinity of the following G.P.s exist, if exists find them:
9, 8.1, 7.29, ...
Find : `sum_("r" = 1)^oo 4(0.5)^"r"`
Find : `sum_("n" = 1)^oo 0.4^"n"`
A ball is dropped from a height of 10m. It bounces to a height of 6m, then 3.6m and so on. Find the total distance travelled by the ball
If the A.M. of two numbers exceeds their G.M. by 2 and their H.M. by `18/5`, find the numbers.
Answer the following:
For a G.P. if t2 = 7, t4 = 1575 find a
Answer the following:
Find k so that k – 1, k, k + 2 are consecutive terms of a G.P.
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 ______.
For a, b, c to be in G.P. the value of `(a - b)/(b - c)` is equal to ______.
The sum of the first three terms of a G.P. is S and their product is 27. Then all such S lie in ______.
For an increasing G.P. a1, a2 , a3 ........., an, if a6 = 4a4, a9 – a7 = 192, then the value of `sum_(i = 1)^∞ 1/a_i` is ______.
If the expansion in powers of x of the function `1/((1 - ax)(1 - bx))` is a0 + a1x + a2x2 + a3x3 ....... then an is ______.
