Equation with integral

Calculus Level 3

For positive integers \(n\geq 2, \) the following equation holds: \[ \int_{0}^{\pi}\sin^n x\ dx = \frac{a_n}{n}\int_{0}^{\pi}\sin^{n-2} x\ dx. \] Find \[\sum_{k=2}^{10} a_k .\]

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