91Ó°ÊÓ

Using induction, verify the inequality. Use the geometric sum to prove that $$ r^{0}+r^{1}+\cdots+r^{n}<\frac{1}{1-r} $$ for all \(n \geq 0\) and \(0

Find the quotient \(q\) and remainder \(r\) as in Theorem 2.5 .6 when \(n\) is divided by \(d\). $$n=-7, d=9$$

Prove that \(X \cap Y \subseteq X\) for all sets \(X\) and \(Y\).

Show, by giving a proof by contradiction, that if 40 coins are distributed among nine bags so that each bag contains at least I coin, at least two bags contain the same number of coins.

Use induction to prove the statement. \(11^{n}-6\) is divisible by \(5,\) for all \(n \geq 1\)

Prove that if \(X \subseteq Y\), then \(X \cap Z \subseteq Y \cap Z\) for all sets \(X, Y\), and \(Z\).

Prove that if \(X \subseteq Y,\) then \(Y-(Y-X)=X\) for all sets \(X\) and \(Y\).

Prove that the number of subsets \(S\) of \(\\{1,2, \ldots, n\\},\) with \(|S|\) even, is \(2^{n-1}, n \geq 1\).

By experimenting with small values of \(n\), guess a formula for the given sum, $$ \frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\cdots+\frac{1}{n(n+1)} $$ then use induction to verify your formula.

Give a tiling of a \(5 \times 5\) board with trominoes in which the upper-left square is missing.