9-4 Sequences & Series

Basic Sequences  Observe patterns!  3, 6, 9, 12, 15  2, 4, 8, 16, 32, …, 2 k, …  {1/k: k = 1, 2, 3, …}  (a 1, a 2, a 3, …, a k, …}  Finite Sequence: has a definite end / last term  Infinite Sequence : continues infinitely

Explicit vs. Recursive Explicit formula: A function used to find the required term. Recursive formula: A function that uses the previous terms to find the required term.

Explicit Sequence Ex: Find the first 6 terms and the 100 th term of the explicitly-defined sequence c n = n 3 – n c1c2c3c1c2c3 c 4 c 5 c 6 c 100

Recursive Sequence Ex: Find the first 4 terms and the 8 th term of the recursively-defined sequence a 1 = 8 and a n = a n-1 – 4, for n ≥ 2 a1a2a3a1a2a3 a4a8a4a8

Arithmetic Sequence The pattern is addition! A sequence {a n } is an arithmetic sequence if it can be written explicitly in the form a n = a 1 + (n – 1)d for some constant d, where d is the common difference (aka pattern number) Each term can be obtained recursively by a n = a n-1 + d (for all n ≥ 2)

Arithmetic Sequence Example Ex: For the arithmetic sequence below, find a)The common difference b)The tenth term c)A recursive rule for the n th term d)An explicit rule for the n th term 6, 10, 14, 18, …

You try! Ex: For the arithmetic sequence below, find a)The common difference b)The tenth term c)A recursive rule for the n th term d)An explicit rule for the n th term 4, 1, -2, -5, …

Geometric Sequence The pattern is multiplication! A sequence {a n } is a geometric sequence if it can be written explicitly in the form a n = a 1 · r n – 1 for some nonzero constant r, where r is the common ratio (aka pattern number) Each term can be obtained recursively by a n = a n-1 · r (for all n ≥ 2)

Geometric Sequence Example Ex: For the geometric sequence below, find a)The common ratio b)The tenth term c)A recursive rule for the n th term d)An explicit rule for the n th term 2, 6, 18, 54, …

You try! Ex: For the geometric sequence below, find a)The common ratio b)The tenth term c)A recursive rule for the n th term d)An explicit rule for the n th term 1, -2, 4, -8, 16, …

Constructing Sequences Ex: The second and fifth terms of a sequence are 6 and 48, respectively. Find explicit and recursive formulas for the sequence if it is a) arithmetic and b) geometric.

Fibonacci Sequence

It’s a race! Who can be the first one to find the sum of all numbers from 1 – 100 ?

Sigma Notation This is a shorthand way to represent a large sum of numbers Uses the capital Greek letter sigma, Σ In summation notation, the sum of the terms of the sequence {a 1, a 2, …, a n } is denoted which is read “the sum of a k from k=1 to n” The variable k is called the index of summation

…Say what?!!?? See if you can determine the number represented by each of the following expressions:

Sum of a Finite Arithmetic Sequence Let {a 1, a 2, a 3, …, a n } be a finite arithmetic sequence with common difference d. Then the sum of the terms of the sequence is Proof is on pg 740 if you’re in the mood for some fun!

Revisit Arithmetic Sequences Remember our example 3, 6, 9, 12, 15? Find the sum for this sequence. Use the formula. What about the sum of numbers 1 – 100?

Sum of a Finite Geometric Sequence Let {a 1, a 2, a 3, …, a n } be a finite geometric sequence with common ratio r ≠1. Then the sum of the terms of the sequence is S = Proof is on pg 742 if you want more fun!

Revisit Geometric Sequences Remember our example 2, 4, 8, 16, 32? Find the sum for this sequence. Use the formula. Find the sum for 42, 7,, …,

Infinite Series: Used when adding an infinite number of terms together Not a true sum; how can you find an answer for infinity? We use a sequence of partial sums and limits to find these infinite sums We can only find the sums if the series converges to a single value. If it diverges, the limit DNE and we have no sum.

Does it converge? For each of the following series, find the first five terms in the sequence of partial sums. Which of the series appear to converge? … 2.1 – – – 6 + …

Sum of an Infinite Geometric Series The geometric series converges if and only if |r| < 1. If it does converge, the sum is S = Try this formula with #1 from the last slide!

One more neat trick… Ex: Express the repeating decimal in fraction form.