a body starting from the rest moves with the acceleration”a”.this acceleration is a=pt where p is a constant.what is the displacement of a particle in a time interval at t=0 to t=t1?

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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?