Inductive Proofs for Divisibility Results
Interactive practice questions
We want to use mathematical induction to prove that $n^4-n$n4−n is divisible by $2$2 for all positive integers $n\ge2$n≥2.
Evaluate $n^4-n$n4−n when $n=2$n=2.
Is $n^4-n$n4−n divisible by $2$2 when $n=2$n=2?
No
Yes
To continue our proof by induction, we now assume that $n^4-n$n4−n is divisible by $2$2 for some positive integer $n=k$n=k where $k\ge2$k≥2.
That is, we assume $\left(\editable{}\right)^4-\editable{}=\editable{}Q$()4−=Q for some integer $Q$Q.
We then aim to use this assumption to prove that $n^4-n$n4−n is divisible by $2$2 when $n=\editable{}$n=.
Using the assumption in part (c), we will now test for divisibility when $n=k+1$n=k+1.
To do so, form an expression for $\left(k+1\right)^4-\left(k+1\right)$(k+1)4−(k+1) in terms of $Q$Q.
From parts (a) and (b) we know that $n^4-n$n4−n is divisible by $2$2 when $n=\editable{}$n=.
From parts (c) and (d) we know that if $n^4-n$n4−n is divisible by $2$2 when $n=\editable{}$n= then it is divisible for $n=\editable{}$n=.
Together, these steps prove that if it works for $n=2$n=2, it also works for $n=3,4,5,6$n=3,4,5,6 etc.
Therefore, by induction, $n^4-n$n4−n is divisible by $2$2 for all positive integers $n\ge2$n≥2.
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