Inductive Proofs for divisibility (2)
Interactive practice questions
We want to use mathematical induction to prove that $7^n-3^n$7n−3n is even for all positive integers $n\ge2$n≥2.
Evaluate $7^n-3^n$7n−3n when $n=2$n=2.
Is $7^n-3^n$7n−3n even when $n=2$n=2?
No
Yes
To continue our proof by induction, we now assume that $7^n-3^n$7n−3n is even for some positive integer $n=k$n=k where $k\ge2$k≥2.
That is, we assume $7^{\editable{}}-3^{\editable{}}=\editable{}Q$7−3=Q for some integer $Q$Q.
We then aim to use this assumption to prove that $7^n-3^n$7n−3n is even 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 $7^{k+1}-3^{k+1}$7k+1−3k+1 in factorised form, in terms of $Q$Q.
From parts (a) and (b) we know that $7^n-3^n$7n−3n is even when $n=\editable{}$n=.
From parts (c) and (d) we know that if $7^n-3^n$7n−3n is even when $n=\editable{}$n= then it is even when $n=\editable{}$n=.
Together, these steps prove that if it works when $n=2$n=2, it also works when $n=3,4,5,6$n=3,4,5,6 etc.
Therefore, by induction, $7^n-3^n$7n−3n is even for all positive integers $n\ge2$n≥2.
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