More concepts on basic arithmetic

e^x = 1 + x/1! + (x^2)/2! + (x^3)/3! + … to infinity

log(1+x) = x – x^2/2 + (x^3)/3 - (x^4)/4 + … to infinity

Sum(n) = [n*(n+1)]/2

Sum(n^2) = [n*(n+1)*(2n+1)]/6

Sum(n^3) = [{n*(n+1)}/2]^2

Sum(odd numbers) = Sum(2*n - 1) = n^2

Sum(even numbers) = Sum(2*n) = n*(n+1)

Last digit of (xyz) ^n:-

       Step 1: n/4 = a*Q + r

       Step 2: if r = 0, last digit = 1 (if z is odd) / last digit = 6 (if z is even)

                     if r = 1, last digit = z.

                     if r = 2, last digit = last digit of z^2

                     if r = 3, last digit = last digit of z^3

       If (x + y) =k, max(x*y) = (k^2)/4

                             min(x + y) = 2*[k^(1/2)]

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Last n-th digit of (a*b*c*…) = Remainder[(a*b*c*….)/10^n] = (a*b*c*…) mod 10^n.

Square of any number can be written in the form (3*n) or (3*n + 1). Also, square of any natural number can be written in the form (4*n) or (4*n + 1).

Last digit of square of natural numbers = 0, 1, 4, 5, 6, 9. 

And the second last digit is always even except when last digit is 6, and, it is 2 when the last digit is 5.