Sequences and Series It’s all in Section 9.4a!!!.

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Presentation transcript:

Sequences and Series It’s all in Section 9.4a!!!

“sequence” will refer to Sequence – an ordered progression of numbers – examples: 1. Finite Sequence 2. Infinite Sequences 3. (unless otherwise specified, the word “sequence” will refer to an infinite sequence) 4. which is sometimes abbreviated Notice: In sequence (2) and (3), we are able to define a rule for the k-th number in the sequence (called the k-th term).

Practice Problems Find the first 6 terms and the 100th term of the sequence in which Note: This is an explicit rule for the k-th term

Practice Problems Another way to define sequences is recursively, where we find each term by relating it to the previous term. Find the first 6 terms and the 100th term for the sequence defined recursively by the following conditions: for all n > 1. The sequence: The pattern???

Definition: Arithmetic Sequence A sequence is an arithmetic sequence if it can be written in the form for some constant d. The number d is called the common difference. Each term in an arithmetic sequence can be obtained recursively from its preceding term by adding d: (for all n > 2).

Practice Problems For each of the following arithmetic sequences, find (a) the common difference, (b) the tenth term, (c) a recursive rule for the n-th term, and (d) an explicit rule for the n-th term. 1. (a) The difference ( d ) between successive terms is 4. (b) (c) (d)

Practice Problems For each of the following arithmetic sequences, find (a) the common difference, (b) the tenth term, (c) a recursive rule for the n-th term, and (d) an explicit rule for the n-th term. 2. (a) The difference ( d ) between successive terms is –3. (b) (c) (d)

Practice Problems For each of the following arithmetic sequences, find (a) the common difference, (b) the tenth term, (c) a recursive rule for the n-th term, and (d) an explicit rule for the n-th term. 3. Is this sequence truly arithmetic??? Difference between successive terms: We do have a common difference!!!

Practice Problems For each of the following arithmetic sequences, find (a) the common difference, (b) the tenth term, (c) a recursive rule for the n-th term, and (d) an explicit rule for the n-th term. 3. (a) The difference ( d ) between successive terms is ln(2). (b)

Practice Problems For each of the following arithmetic sequences, find (a) the common difference, (b) the tenth term, (c) a recursive rule for the n-th term, and (d) an explicit rule for the n-th term. 3. (c) (d)

Geometric Sequences

Definition: Geometric Sequence A sequence is a geometric sequence if it can be written in the form for some non-zero constant r. The number r is called the common ratio. Each term in a geometric sequence can be obtained recursively from its preceding term by multiplying by r : (for all n > 2).

Guided Practice For each of the following geometric sequences, find (a) the common ratio, (b) the tenth term, (c) a recursive rule for the n-th term, and (d) an explicit rule for the n-th term. 1. (a) The ratio ( r ) between successive terms is 2. (b) (c) (d)

Guided Practice For each of the following geometric sequences, find (a) the common ratio, (b) the tenth term, (c) a recursive rule for the n-th term, and (d) an explicit rule for the n-th term. 2. (a) Apply a law of exponents: (b)

Guided Practice For each of the following geometric sequences, find (a) the common ratio, (b) the tenth term, (c) a recursive rule for the n-th term, and (d) an explicit rule for the n-th term. 2. (c) (d)

Guided Practice For each of the following geometric sequences, find (a) the common ratio, (b) the tenth term, (c) a recursive rule for the n-th term, and (d) an explicit rule for the n-th term. 3. (a) The ratio ( r ) between successive terms is –1/2. (b) (c) (d)

Guided Practice The second and fifth terms of a sequence are 3 and 24, respectively. Find explicit and recursive formulas for the sequence if it is (a) arithmetic and (b) geometric. If the sequence is arithmetic: Explicit Rule: Recursive Rule:

Guided Practice The second and fifth terms of a sequence are 3 and 24, respectively. Find explicit and recursive formulas for the sequence if it is (a) arithmetic and (b) geometric. If the sequence is geometric: Explicit Rule: Recursive Rule: