Sequences and Series Explicit, Summative, and Recursive

Sequences A sequence is an ordered list of numbers. The terms of a sequence are referred to in the subscripted form shown below, where the subscript refers to the location (position) of the term in the sequence.

Explicit Formula A formula that allows direct computation of any term for a sequence a 1, a 2, a 3,..., a n,...

Example 1 Solve the first 3 terms of this sequence.

More Examples

Summation Notation stop value summation Index (formula) start value

Rules of Summation Evaluation The summation operator governs everything to its right, up to a natural break point in the expression. Begin by setting the summation index equal to the start value. Then evaluate the algebraic expression governed by the summation sign. Increase the value of the index by 1. Evaluate the expression governed by the summation sign again, and add the result to the previous value. Keep repeating step 3 until the expression has been evaluated and added for the stop value. At that point the evaluation is complete, and you stop.

Evaluating a Simple Summation Expression

Recursive Formula Recursive formula is a formula that is used to determine the next term of a sequence using one or more of the preceding terms. Recursion is the process of choosing a starting term and repeatedly applying the same process to each term to arrive at the following term. Recursion requires that you know the value of the term immediately before the term you are trying to find.

Recursive Formula A recursive formula always has two parts: 1. the starting value for a the recursion equation for a n as a function of a n-1 (the term before it.)

Example 1 Write the first four terms of the sequence:

Example 1 Answer In recursive formulas, each term is used to produce the next term. Follow the movement of the terms throughout the problem. Answer: -4, 1, 6, 11

Example 2 Write the first 5 terms of the sequence

Example 2 Answer Answer: 3, 15, -75, -375, 1875

Arithmetic Sequences If a sequence of values follows a pattern of adding a fixed amount from one term to the next, it is referred to as an arithmetic sequence. The number added to each term is constant (always the same). The fixed amount is called the common difference, d, To find the common difference, subtract the first term from the second term.

Example Find the Common Difference

Examples

Finding any Term of a Sequence where a 1 is the first term of the sequence, d is the common difference, n is the number of the term to find.

Examples Find the 10 th term of the sequence 3, 5, 7, 9,... n = 10; a 1 = 3, d = 2 The tenth term is 21.

Examples

Find the number of terms in the sequence 7, 10, 13,..., 55. a 1 = 7, a n = 55, d = 3. We need to find n. This question makes NO mention of "sum", so avoid that formula.

Examples Insert 3 arithmetic means between 7 and 23. 7, ____, ____, ____, 23 7, 11, 15, 19, 23

Arithmetic Series The sum of the terms of a sequence is called a series.

Find the sum of the sequence To find the sum of a certain number of terms of an arithmetic sequence: where S n is the sum of n terms (n th partial sum), a 1 is the first term, a n is the n th term

Examples

Example A theater has 60 seats in the first row, 68 seats in the second row, 76 seats in the third row, and so on in the same increasing pattern. If the theater has 20 rows of seats, how many seats are in the theater? 60, 68, 76,... We wish to find "the sum" of all of the seats. n = 20, a 1 = 60, d = 8 and we need a 20 for the sum. There are 2720 seats.

Geometric Sequences If a sequence of values follows a pattern of multiplying a fixed amount (not zero) times each term to arrive at the following term, it is referred to as a geometric sequence. The number multiplied each time is constant (always the same). The fixed amount multiplied is called the common ratio, r, referring to the fact that the ratio (fraction) of the second term to the first term yields this common multiple. To find the common ratio, divide the second term by the first term.

Examples of Common Ratios

Example

Any Term of a Geometric Sequence

Example

Continued…. Now, work backward multiplying by 3 (or dividing by 1/3) to find the actual first term. a 1 = 27

Geometric Series

Example

A ball is dropped from a height of 8 feet. The ball bounces to 80% of its previous height with each bounce. How high (to the nearest tenth of a foot) does the ball bounce on the fifth bounce?

Continued…..