I found an exercise in my book that requested from me to proof that all numbers that bigger than $12$ can be represented by:
They requested an induction proof,and i decided to share my answer with you,just to be sure about it.
we will check three base cases,$n=12,13,14$:
$12=3x+7y \Rightarrow x=4,y=0$
$13=3x+7y \Rightarrow x=2 , y=1$
$14=3x+7y \Rightarrow x=0,y=2$
we can assume that:
We know that modulus 3 divide all the numbers to three groups of numbers:
From the base case we can assume:
$∀a ∈ A:∃ n=3x$
$∀b ∈ B:∃ n=3x+7y$
$∀c ∈ C:∃ n=3x+14$
From that we can assume that:
if $n ∈ A$ and $n-1 > 11$ then $n-1 ∈ C$.
if $n ∈ B$ and $n-1 > 11$ then $n-1 ∈ A$.
if $n ∈ C$ and $n-1 > 11$ then $n-1 ∈ B$.
Is there any mistakes?
Is that a legit induction proof?