Advertisements
Advertisements
प्रश्न
Prove that: (91/3 . 91/9 . 91/27 ... ∞) = 3.
Advertisements
उत्तर
\[\text { LHS }= 9^\frac{1}{3} . 9^\frac{1}{9} . 9^\frac{1}{27} . . . \infty \]
\[ = 9^\left( \frac{1}{3} + \frac{1}{9}\frac{1}{27} . \right) \]
\[ = 9^\left\{ \frac{\left( \frac{1}{3} \right)}{\left( 1 - \frac{1}{3} \right)} \right\} \]
\[ = 9^\frac{\left( \frac{1}{3} \right)}{\left( 1 - \frac{1}{3} \right)} \]
\[ = \sqrt{9}\]
\[ = 3 =\text { RHS }\]
APPEARS IN
संबंधित प्रश्न
Find the sum to indicated number of terms in the geometric progressions 1, – a, a2, – a3, ... n terms (if a ≠ – 1).
Find the sum to indicated number of terms in the geometric progressions x3, x5, x7, ... n terms (if x ≠ ± 1).
Evaluate `sum_(k=1)^11 (2+3^k )`
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.
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 S be the sum, P the product and R the sum of reciprocals of n terms in a G.P. Prove that P2Rn = Sn
The 4th term of a G.P. is square of its second term, and the first term is − 3. Find its 7th term.
Find three numbers in G.P. whose product is 729 and the sum of their products in pairs is 819.
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:
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.
If a, b, c are in G.P., prove that:
\[a^2 b^2 c^2 \left( \frac{1}{a^3} + \frac{1}{b^3} + \frac{1}{c^3} \right) = a^3 + b^3 + c^3\]
If a, b, c are in G.P., prove that:
(a + 2b + 2c) (a − 2b + 2c) = a2 + 4c2.
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, d are in G.P., prove that:
(a2 − b2), (b2 − c2), (c2 − d2) are in G.P.
The sum of two numbers is 6 times their geometric means, show that the numbers are in the ratio `(3+2sqrt2):(3-2sqrt2)`.
If the sum of an infinite decreasing G.P. is 3 and the sum of the squares of its term is \[\frac{9}{2}\], then write its first term and common difference.
The value of 91/3 . 91/9 . 91/27 ... upto inf, is
If p, q be two A.M.'s and G be one G.M. between two numbers, then G2 =
If x = (43) (46) (46) (49) .... (43x) = (0.0625)−54, the value of x 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
For the G.P. if a = `7/243`, r = 3 find t6.
For the G.P. if r = − 3 and t6 = 1701, find a.
For what values of x, the terms `4/3`, x, `4/27` are in G.P.?
For the following G.P.s, find Sn.
`sqrt(5)`, −5, `5sqrt(5)`, −25, ...
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)`
The sum of an infinite G.P. is 5 and the sum of the squares of these terms is 15 find the G.P.
Find GM of two positive numbers whose A.M. and H.M. are 75 and 48
Select the correct answer from the given alternative.
The tenth term of the geometric sequence `1/4, (-1)/2, 1, -2,` ... is –
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 sum of the first 5 terms of the G.P. whose first term is 1 and common ratio is `2/3`
Answer the following:
Find three numbers in G.P. such that their sum is 35 and their product is 1000
Answer the following:
For a sequence Sn = 4(7n – 1) verify that the sequence is a G.P.
Answer the following:
Find `sum_("r" = 1)^"n" (2/3)^"r"`
Answer the following:
If p, q, r, s are in G.P., show that (p2 + q2 + r2) (q2 + r2 + s2) = (pq + qr + rs)2
Let A1, A2, A3, .... be an increasing geometric progression of positive real numbers. If A1A3A5A7 = `1/1296` and A2 + A4 = `7/36`, then the value of A6 + A8 + A10 is equal to ______.
