[SOLVED] MATH220 Mathematical Induction

MATH220 Mathematical Induction

Question 1. Prove using mathematical induction that for all n ≥ 1,

Question 2. Use the Principle of Mathematical Induction to verify that, for n any positive integer, 6n − 1 is divisible by 5.

Question 3. Verify that for all n ≥ 1, the sum of the squares of the first 2n positive integers is given by the formula

Question 4. Consider the sequence of real numbers defined by the relations

Use the Principle of Mathematical Induction to show that xn < 4 for all n ≥ 1.

Question 5. Show that n! > 3n for n ≥ 7.

Question 6. Let p0 = 1, p1 = cos θ (for θ some fixed constant) and pn+1 = 2p1pn − pn−1 for n ≥ 1. Use an extended Principle of Mathematical Induction to prove that pn = cos(nθ) for n ≥ 0.

Question 7. Consider the famous Fibonacci sequence defined by the relations x1 = 1, x2 = 1, and xn = xn−1 + xn−2 for n ≥ 3.

(a) Compute x20.

(b) Use an extended Principle of Mathematical Induction in order to show that for n ≥ 1,

(c) Use the result of part (b) to compute x20.