Advertisements
Advertisements
प्रश्न
Which term of the G.P.: `sqrt3, 3, 3sqrt3`, ... is 729?
Which term of the progression `sqrt3, 3, 3sqrt3`, ... is 729?
Advertisements
उत्तर
\[\text { Here, first term, }a = \sqrt{3} \]
\[\text { and common ratio }, r = \sqrt{3}\]
\[\text { Let the } n^{th} \text { term be } 729 . \]
\[ \therefore a_n = 729\]
\[ \Rightarrow a r^{n - 1} = 729\]
\[ \Rightarrow \left( \sqrt{3} \right) \left( \sqrt{3} \right)^{n - 1} = 729\]
\[ \Rightarrow (\sqrt{3} )^{n - 1} = \frac{\left( \sqrt{3} \right)^{12}}{\sqrt{3}} = (\sqrt{3} )^{11} \]
\[ \Rightarrow n - 1 = 11\]
\[ \Rightarrow n = 12\]
\[\text { Thus, the }{12}^{th}\text { term of the given G . P . is } 729 .\]
APPEARS IN
संबंधित प्रश्न
Find the 20th and nthterms of the G.P. `5/2, 5/4 , 5/8,...`
Find the sum to 20 terms in the geometric progression 0.15, 0.015, 0.0015,…
How many terms of G.P. 3, 32, 33, … are needed to give the sum 120?
If the pth, qth and rth terms of a G.P. are a, b and c, respectively. Prove that `a^(q - r) b^(r-p) c^(p-q) = 1`.
If f is a function satisfying f (x +y) = f(x) f(y) for all x, y ∈ N such that f(1) = 3 and `sum_(x = 1)^n` f(x) = 120, find the value of n.
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.
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 12th term of the G.P.
\[\frac{1}{a^3 x^3}, ax, a^5 x^5 , . . .\]
Find :
the 10th term of the G.P.
\[\sqrt{2}, \frac{1}{\sqrt{2}}, \frac{1}{2\sqrt{2}}, . . .\]
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 the G.P.'s 5, 10, 20, ... and 1280, 640, 320, ... have their nth terms equal, find the value of n.
The 4th term of a G.P. is square of its second term, and the first term is − 3. Find its 7th term.
If the pth and qth terms of a G.P. are q and p, respectively, then show that (p + q)th term is \[\left( \frac{q^p}{p^q} \right)^\frac{1}{p - q}\].
Find three numbers in G.P. whose sum is 38 and their product is 1728.
Find the sum of the following geometric progression:
1, −1/2, 1/4, −1/8, ... to 9 terms;
Find the sum of the following serie:
5 + 55 + 555 + ... to n terms;
Find the sum of the following series:
9 + 99 + 999 + ... to n terms;
Find the sum of the following series:
0.5 + 0.55 + 0.555 + ... to n terms.
How many terms of the sequence \[\sqrt{3}, 3, 3\sqrt{3},\] ... must be taken to make the sum \[39 + 13\sqrt{3}\] ?
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`
If Sp denotes the sum of the series 1 + rp + r2p + ... to ∞ and sp the sum of the series 1 − rp + r2p − ... to ∞, prove that Sp + sp = 2 . S2p.
If a, b, c are in G.P., prove that log a, log b, log c are in A.P.
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 (b2 + c2) = c (a2 + b2)
If a, b, c, d are in G.P., prove that:
(a + b + c + d)2 = (a + b)2 + 2 (b + c)2 + (c + d)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), (b − c), (c − a) are in G.P., then prove that (a + b + c)2 = 3 (ab + bc + ca)
If a, b, c are in G.P., then prove that:
If \[\frac{1}{a + b}, \frac{1}{2b}, \frac{1}{b + c}\] are three consecutive terms of an A.P., prove that a, b, c are the three consecutive terms of a G.P.
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.
Find the geometric means of the following pairs of number:
−8 and −2
If pth, qth and rth terms of an A.P. are in G.P., then the common ratio of this G.P. is
The value of 91/3 . 91/9 . 91/27 ... upto inf, is
The two geometric means between the numbers 1 and 64 are
Mark the correct alternative in the following question:
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 p2R3 : S3 is equal to
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
The numbers 3, x, and x + 6 form are in G.P. Find x
The numbers 3, x, and x + 6 form are in G.P. Find nth term
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 3 years.
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 10 years.
For a G.P. if a = 2, r = 3, Sn = 242 find n
For a G.P. sum of first 3 terms is 125 and sum of next 3 terms is 27, find the value of r
Find the sum to n terms of the sequence.
0.2, 0.02, 0.002, ...
Express the following recurring decimal as a rational number:
`2.bar(4)`
Express the following recurring decimal as a rational number:
`2.3bar(5)`
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 perimeters of all the squares
If the A.M. of two numbers exceeds their G.M. by 2 and their H.M. by `18/5`, find the numbers.
Select the correct answer from the given alternative.
The tenth term of the geometric sequence `1/4, (-1)/2, 1, -2,` ... is –
Select the correct answer from the given alternative.
Sum to infinity of a G.P. 5, `-5/2, 5/4, -5/8, 5/16,...` is –
