ab k + k C a k b + k C a k – b + k C a k – b +...+ k C k - ab k + k C k b k + [by actual multiplication] = k C a k + + ( k C + k C ) a k b + ( k C + k C ) a k – b + ... + ( k C k + k C k – ) ab k + k C k b k + [grouping like terms] = k + C a k + + k + C a k b + k + C a k – b +...+ k + C k ab k + k + C k + b k + (by using k + C = , k C r + k C r – = k + C r and k C k = = k + C k + ) Thus, it has been proved that P ( k + ) is true whenever P( k ) is true. Therefore, by principle of mathematical induction, P( n ) is true for every positive integer n . We illustrate this theorem by expanding ( x + ) : ( x + ) = C x + C x .
. = x + x + x + x + x + x + Thus ( x + ) = x + x + x + x + x +