Add up All The Integers Between 1 And 100, Inclusive.

1. First you need to realize that you have to rearrange the terms to pairs up the numbers;

                                 S = 1+2+3+4+5+........+97+98+99+100
                                    = (1+100)+(2+99)+(3+98)+.....+(50+51)
                                    = 101 * 50
                                    = 5050

         Formula :      n(A + B)/2
                     n = Total no. of numbers
                     A = First No.
                     B = Last No.

         So,     n = Total No. = 100,
                   A = First No. = 1,
                   B = Last No. = 100

         Now, According to the formula : 100(1 + 100) / 2
                                                            = 50 * 101  
                                                            = 5050

• Similarly,

                       If we want to add up all the integers between 1 and 101, inclusive.
                   
                          Then,       S = 1+2+3+......+99+100+101
                                          Now, According to the formula : 101(1 + 101) / 2
                                                                                            = 101 * 51
                                                                                            = 5151