Sequence Series Review

Sequence Series Review

Sequences

• A list of numbers or objects in a specific order.

  • Examples:

    • 2, 4, 6, 8

    • -4, 16, -64

  • Can be infinite or finite.

  • Formal Definition: A function whose domain is the set of natural numbers.

    • Example: 2, 4, 8, 16 → $a<em>n = 2n$ ($a</em>n$ = term (range), $n$ = term # (Domain))

Explicit vs. Recursive

• Explicit: A formula that helps you get any term directly.

  • Example: $a_n = 12n + 3$

• Recursive: A formula that uses a previous term to get the next term.

  • Example: $a<em>n = a</em>{n-1} - 7$, where a term needs to be given to start the sequence.

Summation Notation

• Used when representing series (more compact).

• Use the symbol Σ and "k" for "n".

  • Example:

  $\sum_{k=1}^{6} (4+5(k-1))$

  • 6 → ending term #

  • k=1 → 1st term

  • $4+5(k-1)$ equation w/ no $a_n =$

• Use K with Sigma

Arithmetic Sequences

• A list of numbers with a common difference (d) - add or subtract.

• Formula: $a<em>n = a</em>1 + d(n-1)$

  • Example: If $a<em>5 = 25$, $d = -9$, then $a</em>n = 25 - 9(n-1)$

    • Find "$a_1$"

Geometric Sequences

• A sequence with a common ratio (r) - multiply or divide.

• Formula: $a<em>n = a</em>1 ^{n-1}$

  • Example: If $a<em>2 = 3, r = 5$, then $a</em>n = 3 ^{n-1}$

    • Find "r"