Szerkesztő:Wikizoli/próbalap

A direct proof

The following proof was published by János Varga in issue 2 of Matlap in Cluj-Napoca, 2014, p. 48.

1. Let $h\geq 0$. Then the sum of the geometric progression $1,q,q^{2},...,q^{n}$ with the common ratio $q=1+h$ is $\geq n$, since each member is at least 1. From the sum formula of the geometric series, we get that $\displaystyle {{\frac {(1+h)^{n}-1}{(1+h)-1}}\geq n}$, that is $(1+h)^{n}\geq 1+nh.$.

2. Let $-1\leq h<0$. Then $q\leq 1$, so for the above formula we get $\displaystyle {{\frac {(1+h)^{n}-1}{(1+h)-1}}\leq n}$, and since $h<0$, the result is the same: $(1+h)^{n}\geq 1+nh.$.