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
\[\text { Here, first term, }a = \frac{1}{3} \]
\[\text { and common ratio } r = \frac{1}{3}\]
\[\text { Let the } n^{th}\text { term be } \frac{1}{19683} . \]
\[ \therefore a_n = \frac{1}{19683}\]
\[ \Rightarrow a r^{n - 1} = \frac{1}{19683}\]
\[ \Rightarrow \left( \frac{1}{3} \right) \left( \frac{1}{3} \right)^{n - 1} = \frac{1}{19683}\]
\[ \Rightarrow \left( \frac{1}{3} \right)^{n - 1} = \frac{3}{\left( 3 \right)^9} = \left( \frac{1}{3} \right)^8 \]
\[ \Rightarrow n - 1 = 8 \]
\[ \Rightarrow n = 9\]
\[\text { Thus, the } 9^{th} \text { term of the given G . P . is } \frac{1}{19683} .\]
APPEARS IN
संबंधित प्रश्न
Find the sum to indicated number of terms of the geometric progressions `sqrt7, sqrt21,3sqrt7`...n terms.
The sum of first three terms of a G.P. is `39/10` and their product is 1. Find the common ratio and the terms.
Show that the sequence <an>, defined by an = \[\frac{2}{3^n}\], n ϵ N is a G.P.
Find:
the ninth term of the G.P. 1, 4, 16, 64, ...
Find :
nth term of the G.P.
\[\sqrt{3}, \frac{1}{\sqrt{3}}, \frac{1}{3\sqrt{3}}, . . .\]
Find three numbers in G.P. whose sum is 38 and their product is 1728.
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:
2, 6, 18, ... to 7 terms;
Find the sum of the following geometric series:
0.15 + 0.015 + 0.0015 + ... to 8 terms;
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.
A G.P. consists of an even number of terms. If the sum of all the terms is 5 times the sum of the terms occupying the odd places. Find the common ratio of the G.P.
If a, b, c are in G.P., prove that:
a (b2 + c2) = c (a2 + b2)
If a, b, c are in G.P., prove that:
\[\frac{(a + b + c )^2}{a^2 + b^2 + c^2} = \frac{a + b + c}{a - b + c}\]
If a, b, c are in G.P., prove that the following is also in G.P.:
a2 + b2, ab + bc, b2 + c2
If a, b, c are in A.P. and a, b, d are in G.P., show that a, (a − b), (d − c) are in G.P.
If pth, qth and rth terms of an A.P. and G.P. are both a, b and c respectively, show that \[a^{b - c} b^{c - a} c^{a - b} = 1\]
Insert 6 geometric means between 27 and \[\frac{1}{81}\] .
If the fifth term of a G.P. is 2, then write the product of its 9 terms.
If logxa, ax/2 and logb x are in G.P., then write the value of x.
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 x = (43) (46) (46) (49) .... (43x) = (0.0625)−54, the value of x is
For the G.P. if r = − 3 and t6 = 1701, find a.
Mosquitoes are growing at a rate of 10% a year. If there were 200 mosquitoes in the beginning. Write down the number of mosquitoes after n years.
The numbers x − 6, 2x and x2 are in G.P. Find x
Find: `sum_("r" = 1)^10(3 xx 2^"r")`
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, ...`
Determine whether the sum to infinity of the following G.P.s exist, if exists find them:
9, 8.1, 7.29, ...
Express the following recurring decimal as a rational number:
`51.0bar(2)`
Find `sum_("r" = 0)^oo (-8)(-1/2)^"r"`
The midpoints of the sides of a square of side 1 are joined to form a new square. This procedure is repeated indefinitely. Find the sum of the areas of all the squares
Select the correct answer from the given alternative.
The common ratio for the G.P. 0.12, 0.24, 0.48, is –
Answer the following:
In a G.P., the fourth term is 48 and the eighth term is 768. Find the tenth term
Answer the following:
Find three numbers in G.P. such that their sum is 35 and their product is 1000
If a, b, c, d are four distinct positive quantities in G.P., then show that a + d > b + c
If the pth and qth terms of a G.P. are q and p respectively, show that its (p + q)th term is `(q^p/p^q)^(1/(p - q))`
If x, 2y, 3z are in A.P., where the distinct numbers x, y, z are in G.P. then the common ratio of the G.P. is ______.
For an increasing G.P. a1, a2 , a3 ........., an, if a6 = 4a4, a9 – a7 = 192, then the value of `sum_(i = 1)^∞ 1/a_i` is ______.
