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\li Use mathematical induction to prove that the following statement is
true for all positive integers $n.$

$$8+15+22+29+\ldots+(7n+1) = {(7n(n+1)+2n)\over 2}.$$

We start with the base case $n = 1.$
$$8 = {7(1)(1+1)+2(1)\over 2} = {14+2\over 2} = 8.$$

We now inductively assume that the statement is true for some positive
integer $k,$ and prove the statement for $k+1.$

$$8+15+\ldots+(7k+1) = {7k(k+1)+2k\over 2}.$$

We add $7(k+1)+1$ to both sides.
$$8+15+\ldots+(7k+1)+(7(k+1)+1) = {7k(k+1)+2k\over 2} + 7(k+1)+1 =$$$$
{7k(k+1) + 7(2)(k+1) + 2k + 2\over 2} = {7(k+1)(k+2) + 2(k+1)\over 2}.$$

We have thus shown the statement for $k+1.$

By mathematical induction, we have proven the initial statement for all
positive integers $n.$

\li Use mathematical induction to prove or disprove that the following
statement is true for all non-negative integers $n.$

$$1+nh \leq (1+h)^n,\hbox{ where }h>-1.$$

We start with the base case of the smallest non-negative integer $n=0.$
$$1+0h = 1 \leq 1 = (1+h)^0,$$
(note that $h>-1,$ so $(1+h)>0,$ so $(1+h)^0 = 1.$)

We now assume that the statement is true for some non-negative integer
$k$ and prove the statement for $k+1.$

We know the inductive hypothesis $1+kh \leq (1+h)^k$ and multiply both
sides by $1+h$ to get
$(1+kh)(1+h) = 1+(k+1)h+kh^2 \leq (1+h)^{k+1}$
$0\leq h^2,$ so $0\leq kh^2,$ so $1+(k+1)h \leq (1+h)^{k+1}.$

By mathematical induction, we have shown that for all non-negative
integers $n,$ $1+nh \leq (1+h)^n.$

\li Use mathematical induction to prove or disprove that the following
statement is true for all positive odd integers $n.$

$$n^2-1\hbox{ is divisible by }8.$$

We start with the base case of the smallest positive odd integer $n=1.$
$n^2-1 = 1-1 = 0,$ and $8\mid 0,$ showing our base case.

We now assume that the statement is true for some positive odd integer
$k.$ Because $k$ is odd, there is an integer $j$ such that $k = 2j+1.$
We start from the inductive hypothesis $k^2-1$ is divisible by 8, so
there is an $l$ such that $k^2-1 = 8l.$
$$k^2-1 = (2j+1)^2-1 = 4j^2+4j.$$
We add $8j+8$ to both sides.
$$4j^2+12j+8=(2j+3)^2-1 = (k+1)^2-1 = 8(l+j+1),$$
showing that $(k+1)^2-1$ is divisible by 8.

By mathematical induction, we have now shown that, for all positive odd
integers $n,$ $n^2-1$ is divisible by 8.

\li Use mathematical induction to prove that the following statement is
true for all integers $n$ greater than 1.

$$n! < n^n.$$

We start with the base case $n = 2.$ $2! = 2 < 4 = 2^2.$

We now inductively assume the statement is true for some $k \geq 2,$ and
we show that $(k+1)! < (k+1)^{k+1}.$

Noting that $k^k < (k+1)^k$ for $k > 1,$
$$k! < k^k < (k+1)^k.$$

We then multiply both sides by $k+1,$ giving
$$(k+1)! < (k+1)^{k+1}.$$

We have shown by mathematical induction that for all integers $n>1,$
$n! < n^n.$

\bye
