7.1 Sequences and series (in class)

Observe the patterns, write down the next term of the following sequences.

If the general form of the sequence$$\dfrac{1}{1\times2}, \dfrac{1}{2\times3}, \dfrac{1}{3\times4}, \dfrac{1}{4\times5}, \dfrac{1}{5\times6}, \dfrac{1}{6\times7}, \cdots$$ can be written as [math]a_n=\dfrac{1}{f(n)}[/math], what is [math]f(n)[/math]?

Consider the sequence below
$$2, 3, 7, 2, 6, 3, 5, 1, 4, 3, 7, 6, 5, 4$$
For each number [math]n[/math], there are exactly [math]n[/math] numbers between the two [math]n[/math]s. For example, there are 7 numbers between the two 7s; there are 5 numbers between the two 5s. Given four numbers [math]1, 2, 3, 4[/math], construct a sequence that satisfies this special property.

If [math]\{a_n\}=0, 2, 6, 12, 20, 30, 42, 56, 72, \cdots[/math], what is the value of [math]a_{100}[/math]?

If the general form of the sequence$$\dfrac{1}{1}, \dfrac{2}{3}, \dfrac{3}{5}, \dfrac{4}{7}, \dfrac{5}{9}, \dfrac{6}{11}, \cdots$$ can be written as [math]a_n=\dfrac{f(n)}{g(n)}[/math], what are [math]f(n)[/math] and [math]g(n)[/math]?