For and odd integer `n ge 1, n^(3) - (n - 1)^(3) ` + ……
`+ (- 1)^(n-1) 1^(3)`

Explanation

To solve the expression \( n^3 - (n-1)^3 + (n-2)^3 - (n-3)^3 + \ldots + (-1)^{n-1} 1^3 \) for an odd integer \( n \geq 1 \), we can follow these steps:

### Step 1: Understand the Pattern
Since \( n \) is an odd integer, we can denote it as \( n = 2k + 1 \) for some integer \( k \). The series alternates between positive and negative cubes.

### Step 2: Rewrite the Expression
The expression can be rewritten as:
\[
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