7.1 Sequences and series

1. Question

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

(a) [math]3, 6, 12, 24, 48,[/math]Fill in the blank 1 of 4

(b) [math]1, -1, 2, -2, 3,[/math]Fill in the blank 2 of 4

(c) [math]1, 1, 1, 3, 5, 9, 17, 31, 57, 105,[/math]Fill in the blank 3 of 4

(d) (look-and-say sequence) [math]1, 11, 21, 1211, 111221, 312211, 13112221,[/math]Fill in the blank 4 of 4

2. Question

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]?

[math]f(n)=[/math]Fill in the blank 1 of 3 [math]n^2+[/math]Fill in the blank 2 of 3 [math]n+[/math]Fill in the blank 3 of 3

3. Question

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]?

[math]f(n)=[/math]Fill in the blank 1 of 4 [math]n+[/math]Fill in the blank 2 of 4
[math]g(n)=[/math]Fill in the blank 3 of 4 [math]n+[/math]Fill in the blank 4 of 4

4. Question

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.

Fill in the blank 1 of 1

5. Question

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

Fill in the blank 1 of 1

7.1 Sequences and series

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