Bernoullis inequality

Task number: 2798

Show that for all \(n\in \mathbb N\) and \(x\in \mathbb R,\ x> -1\) it is true that \((1+x)^n \ge 1+nx\).

  • Resolution

    For \(n=1\) certainly \(1+x\ge 1+x\).

    Given that the statement is true for \(n\), we show that it holds for \(n+1\):
    \((1+x)^{n+1}=(1+x)^n(1+x)\ge (1+nx)(1+x)=1+(n+1)x+nx^2\ge 1+(n+1)x \).

    Notice that in the first inequality we used the assumption that \(1+x\) is non-negative.

Difficulty level: Easy task (using definitions and simple reasoning)
Proving or derivation task
Cs translation
Send comment on task by email