Advertisements
Advertisements
Question
If a, b, c, d are in G.P, prove that (an + bn), (bn + cn), (cn + dn) are in G.P.
Advertisements
Solution
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
RELATED QUESTIONS
The 5th, 8th and 11th terms of a G.P. are p, q and s, respectively. Show that q2 = ps.
Which term of the following sequence:
`2, 2sqrt2, 4,.... is 128`
Which term of the following sequence:
`sqrt3, 3, 3sqrt3`, .... is 729?
Find the sum to indicated number of terms of the geometric progressions `sqrt7, sqrt21,3sqrt7`...n terms.
Find the sum to indicated number of terms in the geometric progressions 1, – a, a2, – a3, ... n terms (if a ≠ – 1).
Find four numbers forming a geometric progression in which third term is greater than the first term by 9, and the second term is greater than the 4th by 18.
Show that one of the following progression is a G.P. Also, find the common ratio in case:
−2/3, −6, −54, ...
Find :
the 8th term of the G.P. 0.3, 0.06, 0.012, ...
Which term of the G.P. :
\[\frac{1}{3}, \frac{1}{9}, \frac{1}{27} . . \text { . is } \frac{1}{19683} ?\]
The seventh term of a G.P. is 8 times the fourth term and 5th term is 48. Find the G.P.
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:
`3/5 + 4/5^2 + 3/5^3 + 4/5^4 + ....` to 2n terms;
Evaluate the following:
\[\sum^{11}_{n = 1} (2 + 3^n )\]
How many terms of the sequence \[\sqrt{3}, 3, 3\sqrt{3},\] ... must be taken to make the sum \[39 + 13\sqrt{3}\] ?
The 4th and 7th terms of a G.P. are \[\frac{1}{27} \text { and } \frac{1}{729}\] respectively. Find the sum of n terms of the G.P.
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 in G.P. is 56. If we subtract 1, 7, 21 from these numbers in that order, we obtain an A.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
If xa = xb/2 zb/2 = zc, then prove that \[\frac{1}{a}, \frac{1}{b}, \frac{1}{c}\] are in A.P.
Insert 5 geometric means between \[\frac{32}{9}\text{and}\frac{81}{2}\] .
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 p, q be two A.M.'s and G be one G.M. between two numbers, then G2 =
The product (32), (32)1/6 (32)1/36 ... to ∞ is equal to
The numbers 3, x, and x + 6 form are in G.P. Find nth term
The numbers x − 6, 2x and x2 are in G.P. Find x
Find the sum to n terms of the sequence.
0.5, 0.05, 0.005, ...
For a sequence, if Sn = 2(3n –1), find the nth term, hence show that the sequence is a G.P.
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:
`2.3bar(5)`
Insert two numbers between 1 and −27 so that the resulting sequence is a G.P.
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:
Find k so that k – 1, k, k + 2 are consecutive terms of a G.P.
If a, b, c, d are four distinct positive quantities in G.P., then show that a + d > b + c
The third term of G.P. is 4. The product of its first 5 terms is ______.
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 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 ______.
