[ASK] Mathematical Induction

Apr 2018
163
38
Sukoharjo
Prove by mathematical induction that \(\displaystyle 7^n-2^n\) is divisible by 5.

What I've done so far:

For n = 1
\(\displaystyle 7^1-2^1=7-2=5\) (true that it is divisible by 5)
For n = k
\(\displaystyle 7^k-2^k=5a\) (assumed to be true that it is divisible by 5)
For n = k + 1
\(\displaystyle 7^{k+1}-2^{k+1}=7^k\cdot7-2^k\cdot2=7(7^k-2^k)+12\cdot2^k=7(5a)+12\cdot2^k\)
This is where the problem lies. How can I show that \(\displaystyle 12\cdot2^k\) is divisible by 5?
 
Jun 2013
1,112
590
Lebanon
\(\displaystyle 7^{k+1}-2^{k+1}=7\left(7^k-2^k\right)+5\cdot 2^k\)
 
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