Chapter 5.1, Problem 76E

etimes we cannot use mathematical induction to prove a result we believe to be true, but we can use mathematical induction to prove a stronger result. Because the inductive hypothesis of the stronger result provides more to work with, this process is calledinductive loading.We use inductive loading in Exercise 74-76.76.Suppose that we want to prove that

   $\frac{1}{2},\frac{3}{4}\mathrm{...}\frac{2n-1}{2n}<\frac{1}{\sqrt{3n}}$

for all positiveintegers n.

1. Show that if we try to prove this inequality using mathematical induction, the basis step works, but the inductive step fails.

Show that mathematical induction can beused to prove the stronger inequality

   $\frac{1}{2},\frac{3}{4}\mathrm{...}\frac{2n-1}{2n}<\frac{1}{\sqrt{3n+1}}$

for all integersrgreater than 1, which, together with a verification for the case wherer= 1, establishes the weaker inequality we originally tried to prove using mathematical induction.