8.1 Sequences

Quick Review

What you’ll learn about Defining a Sequence Arithmetic and Geometric Sequences Graphing a Sequence Limit of a Sequence Essential Question How can we use calculus to define and evaluate sequences?

Defining a Sequence

Example Defining a Sequence Explicitly 1.Find the first four terms and the 100 th term of the sequence {a n } where Set n equal to 1, 2, 3, 4, and 100.

Example Defining a Sequence Recursively 2.Find the first three terms and the 7 th term of the sequence defined recursively by the conditions: b 1 = 4 and b n = b n – 1 – 2 for all n > 2.

Arithmetic Sequence A sequence {a} is an arithmetic sequence if it can be written in the form {a, a + d, a + 2d,..., a + (n – 1)d,...} for some constant d. The number d is the common difference. Each term in an arithmetic sequence can be obtained recursively from its preceding term by adding d:

Example Defining Arithmetic Sequences 3.Given the arithmetic sequence: – 3, 1, 5, 9,... find a.the common difference, b.the ninth term, c.a recursive rule for the nth term, d.an explicit rule for the nth term.

Geometric Sequence A sequence {a} is an geometric sequence if it can be written in the form {a, a. r, a. r 2,..., a. r n – 1,...} for some nonzero constant r. The number r is the common ratio. Each term in an geometric sequence can be obtained recursively from its preceding term by multiplying by r:

Example Defining Geometric Sequences 4.Given the geometric sequence: 1, – 3, 9, – 27,... find a.the common ratio, b.the tenth term, c.a recursive rule for the nth term, d.an explicit rule for the nth term.

Example Constructing a Sequence 5.The second and fifth term of a geometric sequence are – 6 and 48, respectively. Find the first term, common ratio and an explicit rule for the nth term.

Example Graphing a Sequence Using Parametric Mode 6.Draw a graph of the sequence {a n } with Change the mode on your calculator to parametric and dot. Set your window for the following:

Example Graphing a Sequence Using Sequence Graphing Mode 7.Graph the sequence defined recursively by b 1 = 4 and b n = b n – for all n > 2. Change the mode on your calculator to sequence and dot. Replace b n by u(n). Select nMin = 1, u(n) =u(n – 1) + 2, and u(nMin) = {4}.

Example Graphing a Sequence Using Sequence Graphing Mode 7.Graph the sequence defined recursively by b 1 = 4 and b n = b n – for all n > 2. Set nMin = 1, uMax = 10, PlotStart = 1, PlotStep = 1, and graph in the [0, 10] by [– 5, 25] viewing window.

Limit Let L be a real number. The sequence a has limit L as n approaches ∞ if, given any positive number , there is a positive number M such that for all n > M we have We write and say that the sequence converges to L. Sequences that do not have limits diverge.

Properties of Limits If L and M are real numbers and 1. Sum Rule: andthen 2. Difference Rule: 3. Product Rule: 4. Constant Multiple Rule: 5. Quotient Rule:

Example Finding the Limit of a Sequence 8.Determine whether the sequence converges or diverges. If it converges, find its limit. Graph it, changing the mode to parametric and dot. Find the limit analytically, using the Properties of Limits:

The Sandwich Theorem for Sequences and if there is an integer N for which Absolute Value Theorem

Example Using the Sandwich Theorem to find the Limit of a Sequence 9.Show that the following sequence converges and find its limit.

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