sec mathematical proof techniques that ( ( )( )/ and because (ns( nwe can substitute for ( to get - = +ni= ( )( + ( thusby mathematical inductions(nn ( )/ = note carefully what took place in this example first we cast (nin terms of smaller occurrence of the problems(ns( this is important because once ( comes into the picturewe can use the induction hypothesis to replace ( )with ( )( )/ from hereit is simple algebra to prove that ( equals the right-hand side of the original theorem example here is another simple proof by induction that illustrates choosing the proper variable for induction we wish to prove by induction that the sum of the first positive odd numbers is first we need way to describe the nth odd numberwhich is simply this also allows us to cast the theorem as summation pn theorem = ( proofthe base case of yields which is true the induction hypothesis is - ( ( ) = we now use the induction hypothesis to show that the theorem holds true for the sum of the first odd numbers is simply the sum of the first odd numbers plus the nth odd number in the second line belowwe will use the induction hypothesis to replace the partial summation (shown in brackets in the first linewith its closed-form solution after thatalgebra