Describing Sequences

Consider the sequence $(a_n)_{n \ge 1}$ that starts $1, 3, 5, 7, 9, \ldots$ (i.e., the odd numbers in order).

1. Give a recursive definition and closed formula for the sequence.

2. Write out the sequence $(b_n)_{n \ge 2}$ of partial sums of $(a_n)$. Write down the recursive definition for $(b_n)$ and guess at the closed formula.

Consider the three sequences below. For each, find a recursive definition. How are these sequences related?

1. $2, 4, 6, 10, 16, 26, 42, \ldots$.
2. $5, 6, 11, 17, 28, 45, 73, \ldots$.
3. $0, 0 , 0 , 0 , 0 , 0 , 0 ,\ldots$.

Write out the first few terms of the sequence given by $a_1 = 3$; $a_n = 2a_{n-1} + 4$. Then find a recursive definition for the sequence $10, 24, 52, 108, \ldots$.

Write out the first few terms of the sequence given by $a_n = n^2 - 3n + 1$. Then find a closed formula for the sequence (starting with $a_1$) $0, 2, 6, 12, 20, \ldots$.

Show that $a_n = 3\cdot 2^n + 7\cdot 5^n$ is a solution to the recurrence relation $a_n = 7a_{n-1} - 10a_{n-2}$. What would the initial conditions need to be for this to be the closed formula for the sequence?

Use summation ($\sum$) or product ($\prod$) notation to rewrite the following.

1. $2 + 4 + 6 + 8 + \cdots + 2n$.
2. $1 + 5 + 9 + 13 + \cdots + 425$.
3. $1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots + \frac{1}{50}$.
4. $2 \cdot 4 \cdot 6 \cdot \cdots \cdot 2n$.
5. $(\frac{1}{2})(\frac{2}{3})(\frac{3}{4})\cdots(\frac{100}{101})$.

Expand the following sums and products. That is, write them out the long way.

1. $\d\sum_{k=1}^{100} (3+4k)$.
2. $\d\sum_{k=0}^n 2^k$.
3. $\d\sum_{k=2}^{50}\frac{1}{(k^2 - 1)}$.
4. $\d\prod_{k=2}^{100}\frac{k^2}{(k^2-1)}$.
5. $\d\prod_{k=0}^n (2+3k)$.