Advertisements
Advertisements
प्रश्न
Show that the sequence <an>, defined by an = \[\frac{2}{3^n}\], n ϵ N is a G.P.
Advertisements
उत्तर
We have,
\[ a_n = \frac{2}{3^n}, n \in N\]
\[\text { Putting n } = 1, 2, 3, . . . \]
\[ a_1 = \frac{2}{3^1} = \frac{2}{3}, a_2 = \frac{2}{3^2} = \frac{2}{9}, a_3 = \frac{2}{3^3} = \frac{2}{27} \text { and so on } . \]
\[\text { Now, } \frac{a_2}{a_1} = \frac{\frac{2}{9}}{\frac{2}{3}} = \frac{1}{3}, \frac{a_3}{a_2} = \frac{\frac{2}{27}}{\frac{2}{9}} = \frac{1}{3} \text { and so on } . \]
\[ \therefore \frac{a_2}{a_1} = \frac{a_3}{a_2} = . . . = \frac{1}{3}\]
\[\text { So, the sequence is an G . P . , where } \frac{2}{3} \text { is the first term and } \frac{1}{3}\text { is the common ratio }.\]
APPEARS IN
संबंधित प्रश्न
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… .
Insert two numbers between 3 and 81 so that the resulting sequence is G.P.
Show that one of the following progression is a G.P. Also, find the common ratio in case:
4, −2, 1, −1/2, ...
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 G.P.: `sqrt3, 3, 3sqrt3`, ... is 729?
In a GP the 3rd term is 24 and the 6th term is 192. Find the 10th term.
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 progression:
1, 3, 9, 27, ... to 8 terms;
If a and b are the roots of x2 − 3x + p = 0 and c, d are the roots x2 − 12x + q = 0, where a, b, c, d form a G.P. Prove that (q + p) : (q − p) = 17 : 15.
Find the sum of the following series to infinity:
10 − 9 + 8.1 − 7.29 + ... ∞
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:
\[a^2 b^2 c^2 \left( \frac{1}{a^3} + \frac{1}{b^3} + \frac{1}{c^3} \right) = a^3 + b^3 + c^3\]
If a, b, c, d are in G.P., prove that:
(b + c) (b + d) = (c + a) (c + d)
If a, b, c are in G.P., prove that the following is also in G.P.:
a2, b2, c2
If (a − b), (b − c), (c − a) are in G.P., then prove that (a + b + c)2 = 3 (ab + bc + ca)
If pth, qth and rth terms of an A.P. are in G.P., then the common ratio of this G.P. is
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 ______.
Check whether the following sequence is G.P. If so, write tn.
1, –5, 25, –125 …
Check whether the following sequence is G.P. If so, write tn.
`sqrt(5), 1/sqrt(5), 1/(5sqrt(5)), 1/(25sqrt(5))`, ...
For what values of x, the terms `4/3`, x, `4/27` are in G.P.?
The fifth term of a G.P. is x, eighth term of a G.P. is y and eleventh term of a G.P. is z verify whether y2 = xz
If p, q, r, s are in G.P. show that p + q, q + r, r + s are also in G.P.
For a sequence, if Sn = 2(3n –1), find the nth term, hence show that the sequence is a G.P.
Find : `sum_("r" = 1)^oo (-1/3)^"r"`
A ball is dropped from a height of 10m. It bounces to a height of 6m, then 3.6m and so on. Find the total distance travelled by the ball
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.
The tenth term of the geometric sequence `1/4, (-1)/2, 1, -2,` ... is –
Answer the following:
For a sequence Sn = 4(7n – 1) verify that the sequence is a G.P.
Answer the following:
If for a G.P. first term is (27)2 and seventh term is (8)2, find S8
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
In a G.P. of even number of terms, the sum of all terms is 5 times the sum of the odd terms. The common ratio of the G.P. is ______.
The sum or difference of two G.P.s, is again a G.P.
