If ` G_(1)` is the first of n G.M. s between positive numbers a and b , then show that ` G_(1)^(n+1) = a^(n) b ` .

Explanation

To prove that \( G_1^{n+1} = a^n b \) where \( G_1 \) is the first of \( n \) geometric means between positive numbers \( a \) and \( b \), we can follow these steps:

### Step 1: Understand the arrangement of terms
Let \( G_1, G_2, \ldots, G_n \) be the \( n \) geometric means between \( a \) and \( b \). The sequence of terms can be expressed as:
- First term: \( a \)
- Geometric means: \( G_1, G_2, \ldots, G_n \)
- Last term: \( b \)

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