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Sum of first 'n' number of natural numbers Choose any set of natural numbers starting with '1'. Multiply the last number to the next number. Then divide the product by two. Example: Sum of first 10 natural numbers = [10(10+1)]/2 = (10*11)/2      = 110/2            = 55                Sum of first 100 natural numbers = [100(100+1)]/2 = (100*101)/2      = 10100/2            = 5050                 Proof Sn = n/2(a+l) => Sn = n/2(1+n)     => Sn = n(n+1)/2     where, Sn = Sum                                        n = number of terms              l = last term = n Average of first 'n' number of natural...

Numbers that end in '5' Choose any number that ends in 5. Remove '5' from the number. Multiply the left number to its predecessor (next number). Then write '25'. Example: (45)^2 = Remove 5 from the number. The number left is '4'. Multiply 4 with its next number i.e. 5. and write 25 1. (45)^2       = 4(4+1)|25   = 4(5)|25  = 20|25  = 2025   2. (65)^2        = 6(6+1)|25    = 6(7)|25 = 42|25  = 4225   Proof (10n+5)^2                      = (10n)^2+2*10n*5+(5)^2           = 100n^2+100n+25       = (n^2+n)100+25        = [n(n+1)]100+25  = n(n+1)|25      Numbers near to '50' (25-75) To help you I recommend you to memorize some squares :- 1,2,3,4,5,............,23,24,25,50,100 Choose any number between ...