[问答题]
This question asks you to investig
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b2e8iyyrl 4z y5,em o )yk z67m2(qrshsome properties of hexagonal numbers.
Hexagonal numbers 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}$ .
