Suppose a and b are real numbers and 0 a b Prove that for

Suppose a and b are real numbers and 0 ? a ? b.

Prove that for all n ? P,

Solution

P(2) :(a+b)²/4 -a ²+b² /2 =a²+b²+2ab-2a²-2b² /4 =-(a2+b²-2ab) /4 =(a-b)²/4 less than0 thereforeP(2)is true LetP(m) is true that is(b a+b/2 ) ^miis less than equal toa^m+ b^m/2 hence (a+b/2^m (a+b/2)less than equal to(a+b/2)(a^m+b^m/2)that isless than equal toa^m+1+b^m+1+a^mb+ab^m/4is less than equal toa^m+1+b^m+!/2 as a,bgreater than equal to 0 hence P(n) is true