Advertisements
Advertisements
प्रश्न
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
Advertisements
उत्तर
tn = `(5^("n"-3))/(2^("n"-3)) = (5/2)^("n"-3)`
∴ tn+1 = `(5/2)^("n"+1-3) = (5/2)^("n"-2)`
∴ `("t"_("n"+1))/"t"_"n" = ((5/2)^("n"-2))/((5/2)^("n"-3))`
= `(5/2)^("n" - 2 - "n" + 3)`
= `5/2`, which is a constant
∴ the sequence is a G.P. whose common ratio is `5/2`.
Now, tn = `(5/2)^("n" - 3)`
∴ the first term = t1 = `(5/2)^(1 - 3)`
= `(5/2)^(-2)`
= `(2/5)^2`
= `4/25`
Hence, the first term = t1 = `4/25`
and the common ratio = r = `5/2`.
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,…
Find the sum to n terms of the sequence, 8, 88, 888, 8888… .
If a, b, c and d are in G.P. show that (a2 + b2 + c2) (b2 + c2 + d2) = (ab + bc + cd)2 .
Insert two numbers between 3 and 81 so that the resulting sequence is G.P.
A G.P. consists of an even number of terms. If the sum of all the terms is 5 times the sum of terms occupying odd places, then find its common ratio.
Find:
the ninth term of the G.P. 1, 4, 16, 64, ...
The sum of first three terms of a G.P. is \[\frac{39}{10}\] and their product is 1. Find the common ratio and the terms.
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.
Find the sum of the following geometric series:
`sqrt7, sqrt21, 3sqrt7,...` to n terms
Evaluate the following:
\[\sum^{11}_{n = 1} (2 + 3^n )\]
How many terms of the G.P. 3, 3/2, 3/4, ... be taken together to make \[\frac{3069}{512}\] ?
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.
The fifth term of a G.P. is 81 whereas its second term is 24. Find the series and sum of its first eight terms.
Prove that: (21/4 . 41/8 . 81/16. 161/32 ... ∞) = 2.
Find the rational numbers having the following decimal expansion:
\[0 . 6\overline8\]
Find an infinite G.P. whose first term is 1 and each term is the sum of all the terms which follow it.
Three numbers are in A.P. and their sum is 15. If 1, 3, 9 be added to them respectively, they form a G.P. Find the numbers.
If a, b, c are in G.P., prove that:
\[\frac{1}{a^2 - b^2} + \frac{1}{b^2} = \frac{1}{b^2 - c^2}\]
If a, b, c, d are in G.P., prove that:
(a2 + b2 + c2), (ab + bc + cd), (b2 + c2 + d2) are in G.P.
If a, b, c are in G.P., then prove that:
If a, b, c are in A.P., b,c,d are in G.P. and \[\frac{1}{c}, \frac{1}{d}, \frac{1}{e}\] are in A.P., prove that a, c,e are in G.P.
If a, b, c are three distinct real numbers in G.P. and a + b + c = xb, then prove that either x< −1 or x > 3.
Find the geometric means of the following pairs of number:
2 and 8
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 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
If second term of a G.P. is 2 and the sum of its infinite terms is 8, then its first term is
If p, q be two A.M.'s and G be one G.M. between two numbers, then G2 =
The numbers x − 6, 2x and x2 are in G.P. Find 1st term
If Sn, S2n, S3n are the sum of n, 2n, 3n terms of a G.P. respectively, then verify that Sn (S3n – S2n) = (S2n – Sn)2.
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]
Find GM of two positive numbers whose A.M. and H.M. are 75 and 48
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 four distinct positive quantities in G.P., then show that a + d > b + c
The sum of the first three terms of a G.P. is S and their product is 27. Then all such S lie in ______.
If 0 < x, y, a, b < 1, then the sum of the infinite terms of the series `sqrt(x)(sqrt(a) + sqrt(x)) + sqrt(x)(sqrt(ab) + sqrt(xy)) + sqrt(x)(bsqrt(a) + ysqrt(x)) + ...` is ______.
