Prove that (i-1) ( + 1) = n! − 1, for all integers n ≥ 1. Hint: j! (j-1)! × j when j > 0, and 0! = 1. • To receive credit, mathematical induction must be used (proofs using collapsing sum will not receive credit).

Q: Prove that 1+ 2 + 22 + 23 +. + 2" = 2n+1-1 for n 2 1. Use proof by induction.

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Q: Use the principle of mathematical induction to prove | (In x)" da = (-1)" - n!

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Q: =-rگت( (1+ V5) )(n- 2) < fn (п— 2) Use induction to prove that 2 ((1+ |(n–1). for all n 2 3.

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Q: 1 se mathematical induction to prove that 1 + + > Vn +1 for every integer n > 3. + ... Vn

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Q: Show that 1+2+3+… +n = (";").

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Q: Prove that for all natural numbers k, a (5*2^(2k+1) - 1)/3 + a*4^k is of the form 5 + 6a.

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Q: # 2. Use mathematical induction II to prove that for all integers n 2 5, 1 + 4n < 2".

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Q: A positive integer n is called superperfect if ơ(ơ(n)) = 2n Show that if n = 2q, where 2q+1 - 1 is…

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