Homework Week 6

Official assignment: Do as many as needed from 3-17; do as many as needed from 18-24; do 31, 32, 33, 34.

Problem 3

Prove:

\[ 1^2+2^2+\cdots+n^2=\frac{n(n+1)(2n+1)}{6}. \]

Problem 4

Prove:

\[ 1^3+2^3+\cdots+n^3=\left(\frac{n(n+1)}{2}\right)^2. \]

Problem 18

Prove:

\[ n!<n^n \quad \text{for } n>1. \]

Problem 19

Prove:

\[ 1+\frac14+\frac19+\cdots+\frac{1}{n^2}<2-\frac1n. \]

Problem 21

Prove:

\[ 2^n>n^2 \quad \text{for } n>4. \]

Problem 31

Prove that 2 divides \(n^2+n\) whenever \(n\) is a positive integer.

Problem 32

Prove that 3 divides \(n^3+2n\) whenever \(n\) is a positive integer.

Problems 33 and 34

Exact text not pasted in chat. Listed in the official Week 6 assignment.