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Show that for every n∈N, 3

Prove by Induction.

Attempt)

Base Case: n = 1, 3

So the base case holds true.

Assume that the inequality holds for n = k

3

Show true for n = k+1

3

3

3

81 * 3

(80+1) * 3

80(3

80(3

What next?

^{4n+2}+1 is divisible by 10Prove by Induction.

Attempt)

Base Case: n = 1, 3

^{(4(1)+2)}+ 1 = 730So the base case holds true.

Assume that the inequality holds for n = k

3

^{4k+2}+1 is divisible by 10Show true for n = k+1

3

^{4(k+1)+2}+ 13

^{4k+4+2}+ 13

^{4}* 3^{4k+2}+ 181 * 3

^{4k+2}+ 1(80+1) * 3

^{4k+2}+ 180(3

^{4k+2}) + 3^{4k+2}+ 180(3

^{4k+2}) +*is divisible by 10 according to our induction hypotheses.***3**^{4k+2}+ 1What next?

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