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
Which term of the following sequence:
`2, 2sqrt2, 4,.... is 128`
Evaluate `sum_(k=1)^11 (2+3^k )`
Find the sum of the products of the corresponding terms of the sequences `2, 4, 8, 16, 32 and 128, 32, 8, 2, 1/2`
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.
If the first and the nth term of a G.P. are a ad b, respectively, and if P is the product of n terms, prove that P2 = (ab)n.
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`.
Insert two numbers between 3 and 81 so that the resulting sequence is G.P.
The sum of some terms of G.P. is 315 whose first term and the common ratio are 5 and 2, respectively. Find the last term and the number of terms.
Let S be the sum, P the product and R the sum of reciprocals of n terms in a G.P. Prove that P2Rn = Sn
If a, b, c are in A.P,; b, c, d are in G.P and ` 1/c, 1/d,1/e` are in A.P. prove that a, c, e are in G.P.
Show that one of the following progression is a G.P. Also, find the common ratio in case:1/2, 1/3, 2/9, 4/27, ...
Which term of the progression 0.004, 0.02, 0.1, ... is 12.5?
Which term of the G.P.: `sqrt3, 3, 3sqrt3`, ... is 729?
The fourth term of a G.P. is 27 and the 7th term is 729, find the G.P.
Find the sum of the following geometric series:
\[\frac{2}{9} - \frac{1}{3} + \frac{1}{2} - \frac{3}{4} + . . . \text { to 5 terms };\]
Find the sum of the following geometric series:
`3/5 + 4/5^2 + 3/5^3 + 4/5^4 + ....` to 2n terms;
Find the sum of the following geometric series:
1, −a, a2, −a3, ....to n terms (a ≠ 1)
How many terms of the series 2 + 6 + 18 + ... must be taken to make the sum equal to 728?
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).
How many terms of the G.P. `3, 3/2, 3/4` ..... are needed to give the sum `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 series to infinity:
`1/3+1/5^2 +1/3^3+1/5^4 + 1/3^5 + 1/56+ ...infty`
Find the rational numbers having the following decimal expansion:
\[0 .\overline {231 }\]
If a, b, c are in G.P., prove that:
\[a^2 b^2 c^2 \left( \frac{1}{a^3} + \frac{1}{b^3} + \frac{1}{c^3} \right) = a^3 + b^3 + c^3\]
Find the geometric means of the following pairs of number:
a3b and ab3
If the sum of first two terms of an infinite GP is 1 every term is twice the sum of all the successive terms, then its first term is
In a G.P. if the (m + n)th term is p and (m − n)th term is q, then its mth term is
For the G.P. if a = `7/243`, r = 3 find t6.
If p, q, r, s are in G.P. show that p + q, q + r, r + s are also in G.P.
If one invests Rs. 10,000 in a bank at a rate of interest 8% per annum, how long does it take to double the money by compound interest? [(1.08)5 = 1.47]
Select the correct answer from the given alternative.
The common ratio for the G.P. 0.12, 0.24, 0.48, is –
Select the correct answer from the given alternative.
If for a G.P. `"t"_6/"t"_3 = 1458/54` then r = ?
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:
In a G.P., the fourth term is 48 and the eighth term is 768. Find the tenth term
The third term of a G.P. is 4, the product of the first five terms is ______.
Find a G.P. for which sum of the first two terms is – 4 and the fifth term is 4 times the third term.
If the expansion in powers of x of the function `1/((1 - ax)(1 - bx))` is a0 + a1x + a2x2 + a3x3 ....... then an is ______.
