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 ab–c . bc – a . ca – b = 1
Advertisements
उत्तर
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
संबंधित प्रश्न
Which term of the following sequence:
`1/3, 1/9, 1/27`, ...., is `1/19683`?
For what values of x, the numbers `-2/7, x, -7/2` are in G.P?
Find the sum to 20 terms in the geometric progression 0.15, 0.015, 0.0015,…
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.
If the 4th, 10th and 16th terms of a G.P. are x, y and z, respectively. Prove that x, y, z are in G.P.
Find the sum to n terms of the sequence, 8, 88, 888, 8888… .
Find the 4th term from the end of the G.P.
Which term of the G.P.: `sqrt3, 3, 3sqrt3`, ... is 729?
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 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.
The common ratio of a G.P. is 3 and the last term is 486. If the sum of these terms be 728, find the first term.
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.
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 the 4th, 10th and 16th terms of a G.P. are x, y and z respectively. Prove that x, y, z are in G.P.
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 in an infinite G.P., first term is equal to 10 times the sum of all successive terms, then its common ratio is
The value of 91/3 . 91/9 . 91/27 ... upto inf, 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
The product (32), (32)1/6 (32)1/36 ... to ∞ is equal to
For the G.P. if r = − 3 and t6 = 1701, find a.
If p, q, r, s are in G.P. show that p + q, q + r, r + s are also in G.P.
The numbers 3, x, and x + 6 form are in G.P. Find nth 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 3 years.
For a G.P. if a = 2, r = 3, Sn = 242 find n
For a G.P. If t3 = 20 , t6 = 160 , find S7
Find the sum to n terms of the sequence.
0.5, 0.05, 0.005, ...
The value of a house appreciates 5% per year. How much is the house worth after 6 years if its current worth is ₹ 15 Lac. [Given: (1.05)5 = 1.28, (1.05)6 = 1.34]
Determine whether the sum to infinity of the following G.P.s exist, if exists find them:
`1/2, 1/4, 1/8, 1/16,...`
Determine whether the sum to infinity of the following G.P.s exist, if exists find them:
`2, 4/3, 8/9, 16/27, ...`
Express the following recurring decimal as a rational number:
`0.bar(7)`
The midpoints of the sides of a square of side 1 are joined to form a new square. This procedure is repeated indefinitely. Find the sum of the perimeters of all the squares
Select the correct answer from the given alternative.
Sum to infinity of a G.P. 5, `-5/2, 5/4, -5/8, 5/16,...` is –
Answer the following:
If pth, qth and rth terms of a G.P. are x, y, z respectively. Find the value of xq–r .yr–p .zp–q
At the end of each year the value of a certain machine has depreciated by 20% of its value at the beginning of that year. If its initial value was Rs 1250, find the value at the end of 5 years.
If a, b, c, d are in G.P., prove that a2 – b2, b2 – c2, c2 – d2 are also in G.P.
If a, b, c, d are four distinct positive quantities in G.P., then show that a + d > b + c
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 ______.
