math homework

Homework 10 (Last one)

Homework 10: Due Tuesday December 5

1. Let n be a natural number. Prove that

1 + 4 + 7 + · · · + (3n − 2) = n(3n − 1)

2

2. Let n be a natural number. Prove that

1 + 5 + 9 + · · · + (4n − 3) = 2n2 − n

3. Let n be a natural number. Prove that

13 + 23 + 33 + · · · + n3 = n2(n + 1)2

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4. Let n be a natural number. Prove that 9n − 4n is divisible by 5.

5. Let n be a natural number. Prove that 7n − 5n is even. (Note: this means prove it’s divisible by 2)

6. (Bonus for fun for those taking calculus). Consider the function f(x) = xex. Consider f(n)(x), the nth

derivative of f(x). Find a formula for this derivative. Prove your formula using induction.