Summation of integrals!

Calculus Level pending

For \(a_{k} =\displaystyle \int_{0}^{\frac{\pi}{2}} \sin 2x {(1 - \sin x)}^{\frac{k-1}{2}} \text{dx}\),

\(\displaystyle \sum_{k=3}^{\infty} (k+1)(a_{k} - a_{k+1})\) can be expressed as \(\frac{a}{b}\), where \(a\), and \(b\) are coprime, then find \(a \times b\).

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