[问答题]
This question asks you
+m gej7nhrc: 7k*-kqi tnfz *7 to investigate some propertie
* -nmfi(5e-s g gwx/bls of hexagonal numbers. Hexagonal
l g e(*-bi/-fwg5 mnsxnumbers can be represented by dots as shown below where $h_{n}$ denotes the n th hexagonal number, $n \in \mathbb{N} $.
Note that 6 points are required to create the regular hexagon $h_{2}$ with side of length 1 , while 15 points are required to create the next hexagon $h_{3}$ with side of length 2 , and so on.
1. Write down the value of $h_{5}$ .
2. By examining the pattern, show that $h_{n+1}=h_{n}+4 n+1, n \in \mathbb{N}$ .
3. By expressing $h_{n}$ as a series, show that $h_{n}=2 n^{2}-n, n \in \mathbb{N}$ .
4. Hence, determine whether 2016 is a hexagonal number.
5. Find the least hexagonal number which is greater than 80000 .
6. Consider the statement:
45 is the only hexagonal number which is divisible by 9 .
Show that this statement is false.
Matt claims that given $h_{1}=1$ and $ h_{n+1}=h_{n}+4 n+1$, n $\in \mathbb{N}$ , then
$h_{n}=2 n^{2}-n, \quad n \in \mathbb{N}$
7. Show, by mathematical induction, that Matt's claim is true for all n $\in \mathbb{N}$ .
