Please provide sufficient Explanation.....A ball is projected upwards from a height h above the surface of the earth with velocity v. Find time at which it strike the ground

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Sum of series 2[n/2] ! [n/2]!/ n! [2Co-2C2+3C2....+[-1]^n[n+1]2Cn] where n is an even positive integer. Kindly provide solution of this question and how to generalise this. How do we know that which expansion will give this?

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