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[SOLVED] Proof by mathematical induction
Prove by mathematical induction that for all +ve integers n,[itex]10^{3n}+13^{n+1}[/itex] is divisible by 7.
Assume true for n=N.
[tex]10^{3N}+13^{N+1}=7A[/tex]
Multiply both sides by ([itex]10^3+13[/itex])
[tex](10^{3N}+13^{N+1})(10^3+13)=7A(10^3+13)[/tex]
[tex]10^{3N+3}+ 10^3(13^{N+1})+13(10^{3N})+13^{N+2}=7A(1013)[/tex]
[tex]10^{3N+3}+13^{N+2}=7A(1013)-10^3(13^{N+1})-13(10^{3N})[/tex]
Here is where I am stuck. I need to show that [itex]10^3(13^{N+1})-13(10^{3N})[/itex] is divisible by 7 now.
What I would like to get is that [itex]10^3(13^{N+1})-13(10^{3N})[/itex] can somehow be manipulated into the initial inductive hypothesis and then it will become true for n=N+1. So I need some help.
Homework Statement
Prove by mathematical induction that for all +ve integers n,[itex]10^{3n}+13^{n+1}[/itex] is divisible by 7.
Homework Equations
The Attempt at a Solution
Assume true for n=N.
[tex]10^{3N}+13^{N+1}=7A[/tex]
Multiply both sides by ([itex]10^3+13[/itex])
[tex](10^{3N}+13^{N+1})(10^3+13)=7A(10^3+13)[/tex]
[tex]10^{3N+3}+ 10^3(13^{N+1})+13(10^{3N})+13^{N+2}=7A(1013)[/tex]
[tex]10^{3N+3}+13^{N+2}=7A(1013)-10^3(13^{N+1})-13(10^{3N})[/tex]
Here is where I am stuck. I need to show that [itex]10^3(13^{N+1})-13(10^{3N})[/itex] is divisible by 7 now.
What I would like to get is that [itex]10^3(13^{N+1})-13(10^{3N})[/itex] can somehow be manipulated into the initial inductive hypothesis and then it will become true for n=N+1. So I need some help.