Advertisements
Advertisements
प्रश्न
If a, b, c, d are in G.P, prove that (an + bn), (bn + cn), (cn + dn) are in G.P.
Advertisements
उत्तर
It is given that a, b, c,and d are in G.P.
∴b2 = ac … (1)
c2 = bd … (2)
ad = bc … (3)
It has to be proved that (an + bn), (bn + cn), (cn + dn) are in G.P. i.e.,
(bn + cn)2 = (an + bn) (cn + dn)
Consider L.H.S.
(bn + cn)2 = b2n + 2bncn + c2n
= (b2)n+ 2bncn + (c2) n
= (ac)n + 2bncn + (bd)n [Using (1) and (2)]
= an cn + bncn+ bn cn + bn dn
= an cn + bncn+ an dn + bn dn [Using (3)]
= cn (an + bn) + dn (an + bn)
= (an + bn) (cn + dn)
= R.H.S.
∴ (bn + cn)2 = (an + bn) (cn + dn)
Thus, (an + bn), (bn + cn), and (cn + dn) are in G.P.
APPEARS IN
संबंधित प्रश्न
Find the sum to 20 terms in the geometric progression 0.15, 0.015, 0.0015,…
Find the sum to indicated number of terms in the geometric progressions 1, – a, a2, – a3, ... n terms (if a ≠ – 1).
Find the sum to indicated number of terms in the geometric progressions x3, x5, x7, ... n terms (if x ≠ ± 1).
The sum of first three terms of a G.P. is `39/10` and their product is 1. Find the common ratio and the terms.
If f is a function satisfying f (x +y) = f(x) f(y) for all x, y ∈ N such that f(1) = 3 and `sum_(x = 1)^n` f(x) = 120, find the value of n.
Show that the sequence <an>, defined by an = \[\frac{2}{3^n}\], n ϵ N is a G.P.
Find the 4th term from the end of the G.P.
\[\frac{1}{2}, \frac{1}{6}, \frac{1}{18}, \frac{1}{54}, . . . , \frac{1}{4374}\]
Find three numbers in G.P. whose sum is 65 and whose product is 3375.
Find three numbers in G.P. whose sum is 38 and their product is 1728.
Find the sum of the following geometric series:
\[\sqrt{2} + \frac{1}{\sqrt{2}} + \frac{1}{2\sqrt{2}} + . . .\text { to 8 terms };\]
Find the sum of the following serie:
5 + 55 + 555 + ... to n terms;
The sum of n terms of the G.P. 3, 6, 12, ... is 381. Find the value of n.
If S1, S2, S3 be respectively the sums of n, 2n, 3n terms of a G.P., then prove that \[S_1^2 + S_2^2\] = S1 (S2 + S3).
Prove that: (21/4 . 41/8 . 81/16. 161/32 ... ∞) = 2.
If a, b, c are in G.P., prove that log a, log b, log c are in A.P.
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 the following is also in G.P.:
a2 + b2, ab + bc, b2 + c2
Find the geometric means of the following pairs of number:
−8 and −2
If a = 1 + b + b2 + b3 + ... to ∞, then write b in terms of a.
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 ______.
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
Check whether the following sequence is G.P. If so, write tn.
7, 14, 21, 28, …
The numbers 3, x, and x + 6 form are in G.P. Find x
The numbers x − 6, 2x and x2 are in G.P. Find nth term
For a G.P. if S5 = 1023 , r = 4, Find a
For a G.P. sum of first 3 terms is 125 and sum of next 3 terms is 27, find the value of r
Determine whether the sum to infinity of the following G.P.s exist, if exists find them:
`-3, 1, (-1)/3, 1/9, ...`
Express the following recurring decimal as a rational number:
`2.bar(4)`
If the first term of the G.P. is 16 and its sum to infinity is `96/17` find the common ratio.
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 areas of all the squares
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)
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:
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.
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 the sum of an infinite GP a, ar, ar2, ar3, ...... . is 15 and the sum of the squares of its each term is 150, then the sum of ar2, ar4, ar6, .... is ______.
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 ______.
Let A1, A2, A3, .... be an increasing geometric progression of positive real numbers. If A1A3A5A7 = `1/1296` and A2 + A4 = `7/36`, then the value of A6 + A8 + A10 is equal to ______.
