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If alpha (!=1) is a nth root of unity th...

If `alpha (!=1)` is a nth root of unity then `S = 1 + 3alpha+ 5alpha^2 + .......... `upto n terms is equal to

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To solve the problem, we need to find the sum \( S = 1 + 3\alpha + 5\alpha^2 + \ldots \) up to \( n \) terms, where \( \alpha \) is an \( n \)th root of unity (and \( \alpha \neq 1 \)). ### Step-by-Step Solution: 1. **Identify the Series**: The series can be expressed as: \[ S = \sum_{k=0}^{n-1} (2k + 1) \alpha^k ...
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