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Question: prove each of the statements using mathematical induction everything except...

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Prove each of the statements using Mathematical Induction

everything except number 17

—6*7^N‐2*3^N is divisible by 4, for N>=1

--For any positive integer number n, n3+ 2 n is divisible by 3

---1+3+5+(2n–1)=n2

--Prove that n! > 2n for n a positive integer greater than or equal to 4.

--A 2n x 2n L‐shape, n >= 0, is a figure of the following form with no missing squares.

Use mathematical induction to prove that a 2n x 2n L‐shape can be tiled with trominoes (i.e. a 2 x 2 L‐shape)

--Provethat Tn =2n –1 forall n>=1 incaseitisknownthat T1 =1 and Tn =2Tn‐1+1.

14. 6x7N - 2x3N is divisible by 4, for N 2 1 15 For any positive integer number n, n3+2 n is divisible by 3 16. 135+ (2n-1)-2 17. t is (incorrectly) proposed that 2+4+.+ (2 x N) - (N 2)(N-1) for N2 2 Show that the Inductive Step is satisfied but that the Basis Step fails. (Hence, we have reached no conclusion for the proposition regardless of N) Prove that n! > 2n for n a positive integer greater than or equal to 4. A 2 x 2 L-shape, n 2 0, is a figure of the following form with no missing squares. Use mathematical induction to prove that a 2 x 2 L-shape can be tiled with 18 19. trominoes (i.e. a 2 x 2 L-shape) 2nx2n 20. Prove that Tn 2 1 for all n2 1 in case it is known that T1 1 and Tn 2 Th-1+ 1.

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