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
Which term of the following sequence:
`2, 2sqrt2, 4,.... is 128`
Find the sum to indicated number of terms in the geometric progressions x3, x5, x7, ... n terms (if x ≠ ± 1).
The sum of two numbers is 6 times their geometric mean, show that numbers are in the ratio `(3 + 2sqrt2) ":" (3 - 2sqrt2)`.
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.
If a, b, c, d are in G.P, prove that (an + bn), (bn + cn), (cn + dn) are in G.P.
Find:
the ninth term of the G.P. 1, 4, 16, 64, ...
Find:
the 10th term of the G.P.
\[- \frac{3}{4}, \frac{1}{2}, - \frac{1}{3}, \frac{2}{9}, . . .\]
Which term of the G.P. :
\[\sqrt{2}, \frac{1}{\sqrt{2}}, \frac{1}{2\sqrt{2}}, \frac{1}{4\sqrt{2}}, . . . \text { is }\frac{1}{512\sqrt{2}}?\]
Which term of the G.P. :
\[\frac{1}{3}, \frac{1}{9}, \frac{1}{27} . . \text { . is } \frac{1}{19683} ?\]
The fourth term of a G.P. is 27 and the 7th term is 729, find the G.P.
The product of three numbers in G.P. is 125 and the sum of their products taken in pairs is \[87\frac{1}{2}\] . Find them.
The sum of three numbers in G.P. is 14. If the first two terms are each increased by 1 and the third term decreased by 1, the resulting numbers are in A.P. Find the numbers.
Find the sum of the following geometric series:
0.15 + 0.015 + 0.0015 + ... to 8 terms;
Find the sum of the following geometric series:
(x +y) + (x2 + xy + y2) + (x3 + x2y + xy2 + y3) + ... to n terms;
Find the sum of the following geometric series:
1, −a, a2, −a3, ....to n terms (a ≠ 1)
Evaluate the following:
\[\sum^{11}_{n = 1} (2 + 3^n )\]
Find the sum of the following series:
0.5 + 0.55 + 0.555 + ... to n terms.
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.
Find the sum of the following serie to infinity:
`2/5 + 3/5^2 +2/5^3 + 3/5^4 + ... ∞.`
Find the rational numbers having the following decimal expansion:
\[0 . \overline3\]
Find an infinite G.P. whose first term is 1 and each term is the sum of all the terms which follow it.
The sum of three numbers a, b, c in A.P. is 18. If a and b are each increased by 4 and c is increased by 36, the new numbers form a G.P. Find 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 pth, qth and rth terms of a G.P. re x, y, z respectively, then write the value of xq − r yr − pzp − q.
The value of 91/3 . 91/9 . 91/27 ... upto inf, is
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
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
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.
For a G.P. If t4 = 16, t9 = 512, find S10
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]
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,...`
Find `sum_("r" = 0)^oo (-8)(-1/2)^"r"`
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 –
Answer the following:
If pth, qth and rth terms of a G.P. are x, y, z respectively. Find the value of xq–r .yr–p .zp–q
If a, b, c, d are four distinct positive quantities in G.P., then show that a + d > b + c
Find a G.P. for which sum of the first two terms is – 4 and the fifth term is 4 times the third term.
Let `{a_n}_(n = 0)^∞` be a sequence such that a0 = a1 = 0 and an+2 = 2an+1 – an + 1 for all n ≥ 0. Then, `sum_(n = 2)^∞ a^n/7^n` is equal to ______.
