Prove using mathematical induction: 3.If n is a counting number then 6 divides n^3 - n....

Prove using mathematical induction: 3.If n is a counting number then 6 divides n^3 - n. 4.The sum of any three consecutive perfect cubes is divisible by 9. 5.The sum of the first n perfect squares is: n(n +1)(2n +1)/ 6

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Colby Messinger answered 6 months ago

5. Without using the method of mathematical induction, prove that 5^n − 3^n + 2n is...

5. Without using the method of mathematical induction, prove that 5^n − 3^n + 2n is divisible by 4 for all natural n.

Prove using the principle of mathematical induction: (i) The number of diagonals of a convex polygon...

Prove using the principle of mathematical induction: (i) The number of diagonals of a convex polygon with n vertices is n(n − 3)/2, for n ≥ 4, (ii) 2n < n! for all n > k > 0, discover the value of k before doing induction

Prove that 3 divides n^3 −n for all n ≥ 1.

prove by using induction. Prove by using induction. If r is a real number with r...

prove by using induction. Prove by using induction. If r is a real number with r not equal to 1, then for all n that are integers with n greater than or equal to one, r + r^2 + ....+ r^n = r(1-r^n)/(1-r)

2. [6 marks] (Induction) Prove that 21 divides 4n+1 + 5 2n−1 whenever n is a...

2. [6 marks] (Induction) Prove that 21 divides 4n+1 + 5 2n−1 whenever n is a positive integer. HINT: 25 ≡ 4(mod 21)

Use a direct proof to prove that 6 divides (n^3)-n whenever n is a non-negative integer.

12 pts) Use Mathematical Induction to prove that an=n3+5n is divisible by 6 when ever n≥0....

12 pts) Use Mathematical Induction to prove that an=n3+5n is divisible by 6 when ever n≥0. You may explicitly use without proof the fact that the product n(n+1) of consecutive integers n and n+1 is always even, that is, you must state where you use this fact in your proof.Write in complete sentences since this is an induction proof and not just a calculation. Hint:Look up Pascal’s triangle. (a) Verify the initial case n= 0. (b) State the induction hypothesis....

a. Use mathematical induction to prove that for any positive integer ?, 3 divide ?^3 +...

a. Use mathematical induction to prove that for any positive integer ?, 3 divide ?^3 + 2? (leaving no remainder). Hint: you may want to use the formula: (? + ?)^3= ?^3 + 3?^2 * b + 3??^2 + ?^3. b. Use strong induction to prove that any positive integer ? (? ≥ 2) can be written as a product of primes.

Prove that the proof by mathematical induction and the proof by strong induction are equivalent

Prove these scenarios by mathematical induction: (1) Prove n2 < 2n for all integers n>4 (2)...

Prove these scenarios by mathematical induction: (1) Prove n2 < 2n for all integers n>4 (2) Prove that a finite set with n elements has 2n subsets (3) Prove that every amount of postage of 12 cents or more can be formed using just 4-cent and 5-cent stamps

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