Prove that $P(1+i)^n \geq P(1+ni)$

Question

I'm trying to prove that compound interest $\geq$ simple interest I am however stuck below:

$F_{compound} \geq F{simple}$ where F is future worth

$P(1+i)^n \geq P(1+ni)$

$(1+i)^n \geq (1+ni)$

I tried using induction but also stuck due to the 2 variables.

Answer 1

Hint: Use the binomial theorem:

$$(a+b)^n=\sum_{x=0}^n \binom{n}{x}\cdot a^{n-x}\cdot b^x$$

In your case $a=1, b=i$. On the RHS evaluate the first two summands, namely for $x=0$ and $x=1$.