Prove by induction that for all natural number \( n \) \[ \sin \alp...

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Prove by induction that for all natural number \( n \)
\[
\sin \alpha+\sin (\alpha+\beta)+\sin (\alpha+2 \beta)+\ldots+\sin (\alpha+\overline{n-1} \beta)=\frac{\sin \left(\alpha+\frac{n-1}{2} \beta\right) \sin \left(\frac{n \beta}{2}\right)}{\sin \left(\frac{\beta}{2}\right)}
\]
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