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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.

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Sequences and series

Sequences and Series

Explicit, Summative, and Recursive


Sequences
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
Explicit Formula

  • A formula that allows direct computation of any term for a sequence a1, a2, a3, . . . , an, . . .


Example 1
Example 1

  • Solve the first 3 terms of this sequence.


More examples
More Examples

  • Find the first 4 terms of:

  • Find the first 4 terms of:

  • Find the first 5 terms of:

  • Find the indicated term of the following:


Summation notation
Summation Notation

stop value

summation

Index

(formula)

start value


Rules of summation evaluation
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
Evaluating a Simple Summation Expression

  • Suppose our list has just 5 numbers, and they are 1,2, 3, 4, and 5. Evaluate

    • Answer:


Evaluating a simple summation expression1
Evaluating a Simple Summation Expression

  • Order of evaluation can be crucial. Evaluate

    • Answer:


Recursive formula
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 formula1
Recursive Formula

  • A recursive formula always has two parts:  1.  the starting value for a1.  2.  the recursion equation for an as a function of an-1 (the term before it.)


Example 11
Example 1

  • Write the first four terms of the sequence:


Example 1 answer
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
Example 2

  • Write the first 5 terms of the sequence


Example 2 answer
Example 2 Answer

  • Answer:  3, 15, -75, -375, 1875


Arithmetic sequences
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
ExampleFind the Common Difference

  • 1, 4, 7, 10, 13, 16

    d = 3

  • 15, 10, 5, 0, -5, -10,

    d = -5

    d = -1/2


Examples
Examples

  • Find the common difference for the arithmetic sequence whose formula is

    an= 6n + 3

  • Hint: Plug in

  • Answer: 6


Finding any term of a sequence
Finding any Term of a Sequence

  • where a1 is the first term of the sequence,d is the common difference, n is the number of the term to find.


Examples1
Examples

  • Find the 10th term of the sequence                          3, 5, 7, 9, ...

  • n = 10;  a1 = 3, d = 2  

  • The tenth term is 21.


Examples2
Examples

  • Find a formula for the sequence                          1, 3, 5, 7, ...

  • Hint: Work the sequence formula backwards

  • Answer


Examples3
Examples

  • Find the number of terms in the sequence                7, 10, 13, ..., 55.  

  • a1 = 7, an = 55,  d = 3.  We need to find n.This question makes NO mention of "sum", so avoid that formula.


Examples4
Examples

  • Insert 3 arithmetic means between 7 and 23.

  • 7, ____, ____, ____, 23

  • 7, 11, 15, 19, 23


Arithmetic series
Arithmetic Series

  • The sum of the terms of a sequence is called a series.


Find the sum of the sequence
Find the sum of the sequence

  • To find the sum of a certain number of terms of an arithmetic sequence:

  • where Sn is the sum of n terms (nth partial sum),a1 is the first term,  an is the nth term


Examples5
Examples

  • Find the sum of the first 12 positive even      integers.

  • Hint:The word "sum" indicates the need for the sum formula.

  • positive even integers:  2, 4, 6, 8, ...     n = 12;  a1 = 2, d = 2

  • We are missing a12, for the sum formula, so we use the "any term" formula to find it.


Example
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,  a1 = 60,  d = 8 and we need a20 for the sum.

There are 2720 seats.


Geometric sequences
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
Examples of Common Ratios

  • 5, 10, 20, 40, ...

    r = 2

  • -11, 22, -44, 88, ...

    r = -2


Example1
Example

Find the first 5 terms given the following

3, 6, 12, 24, 48

1, -0.5, 0.25, -0.125, 0.0625


Any term of a geometric sequence
Any Term of a Geometric Sequence

  • To find any term of a geometric sequence:


Example2
Example

  • Find the 12th term of the geometric sequence

    5, 15, 45, ……..

    = 885,735


Example3
Example

  • Find for the sequence:

    0.5, 3.5, 24.5, 171.5

    n = 8, r = 7,


Example4
Example

  • The third term of a geometric sequence is 3 and the sixth term is 1/9.  Find the first term.

  • Use as the first term.

  • ___ , ___ , _3_ , ___ , ___ , _1/9_

  • Therefore, n = 4 for solving this problem.


Continued
Continued….

  • Now, work backward multiplying by 3 (or dividing by 1/3) to find the actual first term.

    a1= 27


Geometric series
Geometric Series

  • To find the sum of a certain number of terms of a geometric sequence:

    ; where

    where Sn is the sum of n terms (nth partial sum), a1 is the first term,  r is the common ratio


Example5
Example

  • Evaluate using a formula:


Example6
Example

  • Evaluate the sum


Example7
Example

  • Find the sum of the first 8 terms of the     sequence                     -5, 15, -45, 135, ...

    n= 8;  a1 = -5, r= -3


Example8
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?


Continued1
Continued…..

  • Set up a model drawing for each "bounce".                6.4, 5.12, ___, ___, ___  The common ratio is 0.8.

  • The ball will bounce approximately 2.6 feet