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odolwa99
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Can anyone help me confirm if I have solved this correctly?
Many thanks.
Q. [itex]9^n-5^n[/itex] is divisible by 4, for [itex]n\in\mathbb{N}_0[/itex]
Step 1: For [itex]n=1[/itex]...
[itex]9^1-5^1=4[/itex], which can be divided by [itex]4[/itex].
Therefore, [itex]n=1[/itex] is true...
Step 2: For [itex]n=k[/itex]...
Assume [itex]9^k-5^k=4\mathbb{Z}[/itex], where [itex]\mathbb{Z}[/itex] is an integer...1
Show that [itex]n=k+1[/itex] is true...
i.e. [itex]9^{k+1}-5^{k+1}[/itex] can be divided by [itex]4[/itex]
[itex]9^{k+1}-5^{k+1}[/itex] => [itex]9^{k+1}-9\cdot5^k+9\cdot5^k-5^{k+1}[/itex] => [itex]9(9^k-5^k)+5^k(9-5)[/itex] => [itex]9(4\mathbb{Z})+5^k(4)[/itex]...from 1 above => [itex]36\mathbb{Z}+5^k\cdot4[/itex] => [itex]4(9\mathbb{Z}+5^k)[/itex]
Thus, assuming [itex]n=k[/itex], we can say [itex]n=k+1[/itex] is true & true for [itex]n=2,3,[/itex]... & all [itex]n\in\mathbb{N}_0[/itex]
Many thanks.
Homework Statement
Q. [itex]9^n-5^n[/itex] is divisible by 4, for [itex]n\in\mathbb{N}_0[/itex]
The Attempt at a Solution
Step 1: For [itex]n=1[/itex]...
[itex]9^1-5^1=4[/itex], which can be divided by [itex]4[/itex].
Therefore, [itex]n=1[/itex] is true...
Step 2: For [itex]n=k[/itex]...
Assume [itex]9^k-5^k=4\mathbb{Z}[/itex], where [itex]\mathbb{Z}[/itex] is an integer...1
Show that [itex]n=k+1[/itex] is true...
i.e. [itex]9^{k+1}-5^{k+1}[/itex] can be divided by [itex]4[/itex]
[itex]9^{k+1}-5^{k+1}[/itex] => [itex]9^{k+1}-9\cdot5^k+9\cdot5^k-5^{k+1}[/itex] => [itex]9(9^k-5^k)+5^k(9-5)[/itex] => [itex]9(4\mathbb{Z})+5^k(4)[/itex]...from 1 above => [itex]36\mathbb{Z}+5^k\cdot4[/itex] => [itex]4(9\mathbb{Z}+5^k)[/itex]
Thus, assuming [itex]n=k[/itex], we can say [itex]n=k+1[/itex] is true & true for [itex]n=2,3,[/itex]... & all [itex]n\in\mathbb{N}_0[/itex]