Advertisements
Advertisements
प्रश्न
The value of 91/3 . 91/9 . 91/27 ... upto inf, is
पर्याय
(a) 1
(b) 3
(c) 9
(d) none of these
Advertisements
उत्तर
\[(b) 3\]
\[ 9^\frac{1}{3} \times 9^\frac{1}{9} \times 9^\frac{1}{27} \times . . . \infty \]
\[ = 9^\left( \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + . . . \infty \right) \]
\[\text{ Here, it is a G . P . with } a = \frac{1}{3}\text{ and } r = \frac{1}{3} . \]
\[ \therefore 9^\left( \frac{\frac{1}{3}}{1 - \frac{1}{3}} \right) \]
\[ = 9^\left( \frac{1}{2} \right) = 3\]
\[\]
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 1, – a, a2, – a3, ... n terms (if a ≠ – 1).
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.
If the first and the nth term of a G.P. are a ad b, respectively, and if P is the product of n terms, prove that P2 = (ab)n.
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 `1/r^n`.
Show that one of the following progression is a G.P. Also, find the common ratio in case:
−2/3, −6, −54, ...
Find :
nth term of the G.P.
\[\sqrt{3}, \frac{1}{\sqrt{3}}, \frac{1}{3\sqrt{3}}, . . .\]
Which term of the progression 18, −12, 8, ... is \[\frac{512}{729}\] ?
Find the sum of the following geometric progression:
2, 6, 18, ... to 7 terms;
Find the sum of the following geometric series:
\[\frac{a}{1 + i} + \frac{a}{(1 + i )^2} + \frac{a}{(1 + i )^3} + . . . + \frac{a}{(1 + i )^n} .\]
Find the sum of the following geometric series:
x3, x5, x7, ... to n terms
How many terms of the sequence \[\sqrt{3}, 3, 3\sqrt{3},\] ... must be taken to make the sum \[39 + 13\sqrt{3}\] ?
Let an be the nth term of the G.P. of positive numbers.
Let \[\sum^{100}_{n = 1} a_{2n} = \alpha \text { and } \sum^{100}_{n = 1} a_{2n - 1} = \beta,\] such that α ≠ β. Prove that the common ratio of the G.P. is α/β.
Find the sum of 2n terms of the series whose every even term is 'a' times the term before it and every odd term is 'c' times the term before it, the first term being unity.
Express the recurring decimal 0.125125125 ... as a rational number.
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.
Find the geometric means of the following pairs of number:
2 and 8
If a, b, c are in G.P. and x, y are AM's between a, b and b,c respectively, then
If A be one A.M. and p, q be two G.M.'s between two numbers, then 2 A is equal to
Check whether the following sequence is G.P. If so, write tn.
1, –5, 25, –125 …
For what values of x, the terms `4/3`, x, `4/27` are in G.P.?
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
Find the sum to n terms of the sequence.
0.2, 0.02, 0.002, ...
Find: `sum_("r" = 1)^10(3 xx 2^"r")`
Find: `sum_("r" = 1)^10 5 xx 3^"r"`
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]
Express the following recurring decimal as a rational number:
`0.bar(7)`
If the first term of the G.P. is 16 and its sum to infinity is `96/17` find the common ratio.
Find `sum_("r" = 0)^oo (-8)(-1/2)^"r"`
If the A.M. of two numbers exceeds their G.M. by 2 and their H.M. by `18/5`, find the numbers.
The sum of 3 terms of a G.P. is `21/4` and their product is 1 then the common ratio is ______.
Answer the following:
Find the nth term of the sequence 0.6, 0.66, 0.666, 0.6666, ...
Answer the following:
For a G.P. if t2 = 7, t4 = 1575 find a
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 pth, qth, and rth terms of an A.P. and G.P. are both a, b and c respectively, show that ab–c . bc – a . ca – b = 1
The lengths of three unequal edges of a rectangular solid block are in G.P. The volume of the block is 216 cm3 and the total surface area is 252cm2. The length of the longest edge is ______.
The sum of the infinite series `1 + 5/6 + 12/6^2 + 22/6^3 + 35/6^4 + 51/6^5 + 70/6^6 + ....` is equal to ______.
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 ______.
