Quandaries
and Queries 

I am a high school teacher (912) and have this question about induction. Harmonic numbers are H_{n} = 1 + 1/2 + 1/3 + . . . + 1/n Use induction to prove the following theorem: For all natural numbers n, H_{1} + H_{2} + . . . + H_{n} = (1+n)H_{n}  n Thank you Becky 

Hi Becky, First verify that the expression is true when n=1. Notice that the defining relation, H_{n} = 1 + 1/2 + 1/3 + . . . + 1/n, can be written H_{n+1}=H_{n}+1/(n+1), or equivalently H_{n}=H_{n+1}1/(n+1). For the induction step, assume that H_{1} + H_{2} + . . . + H_{n} = (1+n)H_{n}  n is true for n=1,2,...,k. We need to show that H_{1} + H_{2} + . . . + H_{k+1} = (2+k)H_{k}  ( k+1).
Cheers, 

