Home
Class 12
MATHS
Geometric mean of 2, 2^(2), 2^(3),…,2^(n...

Geometric mean of `2, 2^(2), 2^(3),…,2^(n)` is

A

2

B

`2^((n)/(2))`

C

`2^((n+1)/(2))`

D

`2^((n(n+1))/(2))`

Text Solution

AI Generated Solution

The correct Answer is:
To find the geometric mean of the sequence \(2, 2^2, 2^3, \ldots, 2^n\), we will follow these steps: ### Step 1: Identify the terms The terms of the sequence are \(2^1, 2^2, 2^3, \ldots, 2^n\). ### Step 2: Count the number of terms The total number of terms in this sequence is \(n\). ### Step 3: Write the formula for the geometric mean The geometric mean (GM) of a set of numbers is given by the formula: \[ GM = \left( \prod_{i=1}^{n} x_i \right)^{\frac{1}{n}} \] where \(x_i\) are the terms of the sequence. ### Step 4: Multiply the terms The product of the terms is: \[ 2^1 \times 2^2 \times 2^3 \times \ldots \times 2^n \] This can be rewritten using properties of exponents: \[ = 2^{1 + 2 + 3 + \ldots + n} \] ### Step 5: Calculate the sum of the exponents The sum of the first \(n\) natural numbers is given by the formula: \[ 1 + 2 + 3 + \ldots + n = \frac{n(n + 1)}{2} \] Thus, we can substitute this into our product: \[ = 2^{\frac{n(n + 1)}{2}} \] ### Step 6: Apply the geometric mean formula Now, substituting back into the formula for GM: \[ GM = \left(2^{\frac{n(n + 1)}{2}}\right)^{\frac{1}{n}} = 2^{\frac{n(n + 1)}{2n}} = 2^{\frac{n + 1}{2}} \] ### Final Answer Thus, the geometric mean of the sequence \(2, 2^2, 2^3, \ldots, 2^n\) is: \[ \boxed{2^{\frac{n + 1}{2}}} \]
Promotional Banner

Similar Questions

Explore conceptually related problems

The geometric mean of numbers 7, 7^(2),7^(3),…,7^(n), is

Let a_(1), a_(2) ...be positive real numbers in geometric progression. For n, if A_(n), G_(n), H_(n) are respectively the arithmetic mean, geometric mean and harmonic mean of a_(1), a_(2),..., a_(n) . Then, find an expression for the geometric mean of G_(1), G_(2),...,G_(n) in terms of A_(1), A_(2),...,A_(n), H_(1), H_(2),..., H_(n)

What is the arithmetic mean of (1)/(2), (1)/(3), 2n and m ?

S_(1),S_(2), S_(3),...,S_(n) are sums of n infinite geometric progressions. The first terms of these progressions are 1, 2^(2)-1, 2^(3)-1, ..., 2^(n) – 1 and the common ratios are (1)/(2),(1)/(2^(2)),(1)/(2^(3)), ...., (1)/(2^(n)) . Calculate the value of S_(1), +S_(2),+ ... + S_(n).

Compute the geometric mean of 2, 4, 8.

The geometric mean of observation 2,4,16 and 32 is

The arthmetic mean of the numbers 1,3,3^(2), … ,3^(n-1), is

If M_(g,x) is the geometric mean of N x's and M_(g,y) is the geometric mean of N y's, then the grometric mean M_(g) of the 2N values is

The arithmetic mean of 1,2,3,...n is (a) (n+1)/2 (b) (n-1)/2 (c) n/2 (d) n/2+1

The arithmetic mean of 1,2,3,...n is (a) (n+1)/2 (b) (n-1)/2 (c) n/2 (d) n/2+1