Let there be a GP whose first term is a and the common ratio is r. If A and H are the arithmetic mean and harmonic mean respectively for the first n terms of the G P, AH is equal to

Options

A. `a^(2)r^(n-1)`
B. `ar^(n)`
C. `a^(2)r^(n)`
D. none of these

Explanation

To solve the problem, we need to find the product of the arithmetic mean (A) and the harmonic mean (H) for the first n terms of a geometric progression (GP) with first term \( a \) and common ratio \( r \).

### Step-by-Step Solution:

1. **Identify the terms of the GP:**
The first n terms of the GP are:
\[ a, ar, ar^2, \ldots, ar^{n-1} \]

2. **Calculate the Arithmetic Mean (A):**
The arithmetic mean \( A \) of the first n terms is given by:
\[ A = \frac{a + ar + ar^2 + \ldots + ar^{n-1}}{n} \]
The sum of the first n terms of a GP can be calculated using the formula:
\[ S_n = a \frac{r^n - 1}{r - 1} \quad \text{(for } r \neq 1\text{)} \]
Therefore, the arithmetic mean becomes:
\[ A = \frac{S_n}{n} = \frac{a \frac{r^n - 1}{r - 1}}{n} = \frac{a(r^n - 1)}{n(r - 1)} \]

3. **Calculate the Harmonic Mean (H):**
The harmonic mean \( H \) is defined as:
\[ H = \frac{n}{\frac{1}{a} + \frac{1}{ar} + \frac{1}{ar^2} + \ldots + \frac{1}{ar^{n-1}}} \]
The sum of the reciprocals of the first n terms is:
\[ \frac{1}{a} + \frac{1}{ar} + \frac{1}{ar^2} + \ldots + \frac{1}{ar^{n-1}} = \frac{1}{a} \left(1 + \frac{1}{r} + \frac{1}{r^2} + \ldots + \frac{1}{r^{n-1}}\right) \]
This is also a GP with first term 1 and common ratio \( \frac{1}{r} \). The sum of this series is:
\[ \frac{1 - \left(\frac{1}{r}\right)^n}{1 - \frac{1}{r}} = \frac{r^n - 1}{r^{n-1}(r - 1)} \]
Therefore, we can write:
\[ \frac{1}{H} = \frac{1}{a} \cdot \frac{r^n - 1}{r^{n-1}(r - 1)} \]
Thus:
\[ H = \frac{n a r^{n-1}(r - 1)}{r^n - 1} \]

4. **Calculate the product \( AH \):**
Now, we need to find the product \( AH \):
\[ AH = A \cdot H = \left(\frac{a(r^n - 1)}{n(r - 1)}\right) \cdot \left(\frac{n a r^{n-1}(r - 1)}{r^n - 1}\right) \]
Simplifying this expression:
\[ AH = \frac{a^2 (r^n - 1) r^{n-1} (r - 1)}{(r - 1)(r^n - 1)} = a^2 r^{n-1} \]

### Final Result:
Thus, the product of the arithmetic mean and the harmonic mean for the first n terms of the GP is:
\[ AH = a^2 r^{n-1} \]