Advertisements
Advertisements
प्रश्न
Which term of the G.P. :
\[\frac{1}{3}, \frac{1}{9}, \frac{1}{27} . . \text { . is } \frac{1}{19683} ?\]
Advertisements
उत्तर
\[\text { Here, first term, }a = \frac{1}{3} \]
\[\text { and common ratio } r = \frac{1}{3}\]
\[\text { Let the } n^{th}\text { term be } \frac{1}{19683} . \]
\[ \therefore a_n = \frac{1}{19683}\]
\[ \Rightarrow a r^{n - 1} = \frac{1}{19683}\]
\[ \Rightarrow \left( \frac{1}{3} \right) \left( \frac{1}{3} \right)^{n - 1} = \frac{1}{19683}\]
\[ \Rightarrow \left( \frac{1}{3} \right)^{n - 1} = \frac{3}{\left( 3 \right)^9} = \left( \frac{1}{3} \right)^8 \]
\[ \Rightarrow n - 1 = 8 \]
\[ \Rightarrow n = 9\]
\[\text { Thus, the } 9^{th} \text { term of the given G . P . is } \frac{1}{19683} .\]
संबंधित प्रश्न
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`
If a, b, c and d are in G.P. show that (a2 + b2 + c2) (b2 + c2 + d2) = (ab + bc + cd)2 .
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 8th term of the G.P. 0.3, 0.06, 0.012, ...
Find :
nth term of the G.P.
\[\sqrt{3}, \frac{1}{\sqrt{3}}, \frac{1}{3\sqrt{3}}, . . .\]
Which term of the progression 0.004, 0.02, 0.1, ... is 12.5?
Which term of the G.P. :
\[\sqrt{2}, \frac{1}{\sqrt{2}}, \frac{1}{2\sqrt{2}}, \frac{1}{4\sqrt{2}}, . . . \text { is }\frac{1}{512\sqrt{2}}?\]
If a, b, c, d and p are different real numbers such that:
(a2 + b2 + c2) p2 − 2 (ab + bc + cd) p + (b2 + c2 + d2) ≤ 0, then show that a, b, c and d are in G.P.
If \[\frac{a + bx}{a - bx} = \frac{b + cx}{b - cx} = \frac{c + dx}{c - dx}\] (x ≠ 0), then show that a, b, c and d are in G.P.
Find the sum of the following geometric progression:
1, 3, 9, 27, ... to 8 terms;
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:
`sqrt7, sqrt21, 3sqrt7,...` to n terms
The sum of n terms of the G.P. 3, 6, 12, ... is 381. Find the value of n.
The ratio of the sum of the first three terms to that of the first 6 terms of a G.P. is 125 : 152. Find the common ratio.
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 \[\frac{1}{r^n}\].
How many terms of the G.P. `3, 3/2, 3/4` ..... are needed to give the sum `3069/512`?
Find the sum of the terms of an infinite decreasing G.P. in which all the terms are positive, the first term is 4, and the difference between the third and fifth term is equal to 32/81.
Find k such that k + 9, k − 6 and 4 form three consecutive terms of a G.P.
If a, b, c, d are in G.P., prove that:
\[\frac{ab - cd}{b^2 - c^2} = \frac{a + c}{b}\]
If a, b, c are in G.P., prove that the following is also in G.P.:
a3, b3, c3
If a, b, c, d are in G.P., prove that:
(a2 + b2), (b2 + c2), (c2 + d2) are in G.P.
If a, b, c, d are in G.P., prove that:
(a2 − b2), (b2 − c2), (c2 − d2) are in G.P.
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 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.
Insert 6 geometric means between 27 and \[\frac{1}{81}\] .
Find the geometric means of the following pairs of number:
2 and 8
Write the product of n geometric means between two numbers a and b.
The fractional value of 2.357 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
If p, q, r, s are in G.P. show that p + q, q + r, r + s are also in G.P.
A ball is dropped from a height of 80 ft. The ball is such that it rebounds `(3/4)^"th"` of the height it has fallen. How high does the ball rebound on 6th bounce? How high does the ball rebound on nth bounce?
For a G.P. if a = 2, r = 3, Sn = 242 find n
Express the following recurring decimal as a rational number:
`2.3bar(5)`
Find : `sum_("n" = 1)^oo 0.4^"n"`
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.
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 the nth term of the sequence 0.6, 0.66, 0.666, 0.6666, ...
Let S be the sum, P be the product and R be the sum of the reciprocals of 3 terms of a G.P. Then P2 R3 : S3 is equal to ______.
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 ______.
