What is the geometric mean of the sequence 1, 2, 4, 8, ______, 2ⁿ ?

Concept:

GM or Geometric Mean is the mean value or the central term in the set of numbers in geometric progression.

Geometric mean of a geometric sequence with 'n' terms is computed as n^th root of the product of all the terms in sequence taken together.

Calculations:

Given sequence is 1, 2, 4, 8, ______, 2ⁿ

Rewrite the sequence, 2^0, 2¹, 2^2, ....2ⁿ

GM = \(\rm (2^0.2^1.2^2.2^3....2^n)^{\frac{1}{n+1}}\)

⇒GM = \(\rm (2^{0+1+2+3+....+n})^{\frac{1}{n+1}}\)

⇒GM = \(\rm (2^\frac{n(n+1)}{2})^\frac{1}{n+1}\)

⇒GM = \(\rm 2^\frac{n}{2}\)

Hence, the geometric mean of the sequence 1, 2, 4, 8, ______, 2ⁿ is \(\rm 2^\frac{n}{2}\)