- #1
nistaria
- 8
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Hey guys, this is a problem given to us by our professor in one of the worksheets. I would like an opinion to see if this proof is valid
Use mathematical induction to prove that 3[tex]^{n}[/tex]+7[tex]^{n}[/tex]-2 is divisible by 8 for [tex]\forall[/tex]n[tex]\in[/tex][tex]Z[/tex][tex]^{+}[/tex]
Base step:
n=0
30+70-2=1+1-2=0
0 is divisible by 8 therefore it's true for n=0
Inductive step
Assume that 3k+7k-2 is divisible by 8 is true.
We need to show that 3k+1+7k+1-2 is divisible by 8 as well
3k+1+7k+1-2 = 3*3k+7*7k-2
= 3(3k+7k-2)+4*7k+4
By inductive hypothesis we already know that 3k+7k-2 is divisible by 8 .
Now this is my problem, I'm also trying to show that 4*7k+4 is divisible by 8 as well. This is my attempt at trying to prove it
4*7k+4=4*(7k+1)
7k+1 will always be an even number for k[tex]\ge[/tex]0. Multiplied by 4 this number will always be divisible by 8.
My question is do I still have to prove that 4*(7k+1) is divisible by 8 or is stating that is enough proof?
Thanks for reading
Homework Statement
Use mathematical induction to prove that 3[tex]^{n}[/tex]+7[tex]^{n}[/tex]-2 is divisible by 8 for [tex]\forall[/tex]n[tex]\in[/tex][tex]Z[/tex][tex]^{+}[/tex]
Homework Equations
Base step:
n=0
30+70-2=1+1-2=0
0 is divisible by 8 therefore it's true for n=0
The Attempt at a Solution
Inductive step
Assume that 3k+7k-2 is divisible by 8 is true.
We need to show that 3k+1+7k+1-2 is divisible by 8 as well
3k+1+7k+1-2 = 3*3k+7*7k-2
= 3(3k+7k-2)+4*7k+4
By inductive hypothesis we already know that 3k+7k-2 is divisible by 8 .
Now this is my problem, I'm also trying to show that 4*7k+4 is divisible by 8 as well. This is my attempt at trying to prove it
4*7k+4=4*(7k+1)
7k+1 will always be an even number for k[tex]\ge[/tex]0. Multiplied by 4 this number will always be divisible by 8.
My question is do I still have to prove that 4*(7k+1) is divisible by 8 or is stating that is enough proof?
Thanks for reading