please mathematically prove that interest of given period of time is sum of all interest compounded during that period.

Let for the first year onthe principal amount P , rate r% , I be the interest

So , I = P*r*1/100 = pr/100 , Amount = p + I = p(1+r/100)

 

For the second year P = p(1+r/100) , r = r%

So , Again interest I = p(1+r/100)r/100 , Amount = pr/100 + I = p(1+r/100)(1+r/100) = p(1+r/100)^2

As we go on more n number of years we get

Amount = p(1+r/100)^n which is sum of the principal amount and all interest compounded during that period.