Understanding Geometric Mean
The geometric mean is a measure of central tendency that is particularly useful for sets of positive numbers. It is defined as the nth root of the product of n numbers.
Geometric Mean of P, P^2, P^3, ..., P^n
To find the geometric mean of the series P, P^2, P^3, ..., P^n, we follow these steps:
Step 1: Identify the Numbers
- The series consists of n terms: P, P^2, P^3, ..., P^n.
Step 2: Calculate the Product
- The product of these terms is:
- P * P^2 * P^3 * ... * P^n
- This can be simplified to P^(1+2+3+...+n).
Step 3: Sum of Exponents
- The sum of the first n natural numbers (1 + 2 + 3 + ... + n) is given by the formula:
- n(n + 1)/2.
- Hence, the product becomes:
- P^(n(n + 1)/2).
Step 4: Find the Geometric Mean
- The geometric mean (GM) is the nth root of the product:
- GM = (P^(n(n + 1)/2))^(1/n)
- This simplifies to P^((n + 1)/2).
Conclusion
- Therefore, the geometric mean of P, P^2, P^3, ..., P^n is:
- P^((n + 1)/2).
- This formula indicates that the geometric mean grows exponentially with P and is dependent on the number of terms (n) in the series.