Explicit, Summative, and Recursive 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 a1, a2, a3, . . . , an, . . .

Example 1 Solve the first 3 terms of this sequence.

More Examples Find the first 4 terms of: 𝑎 𝑛 =3𝑛−1 Find the indicated term of the following: 𝑎 𝑛 = 2𝑛−1 3 𝑛 𝑎 3 = 𝑎 5 = 𝑎 11 =

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. Step 1: Begin by setting the summation index equal to the start value. Then evaluate the algebraic expression governed by the summation sign. Step 2: 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. Step 3: Keep repeating step 2 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 Suppose our list has just 5 numbers, and they are 1,2, 3, 4, and 5. Evaluate Answer: 1 2 + 2 2 + 3 2 + 4 2 + 5 2 =55

Evaluating a Simple Summation Expression Order of evaluation can be crucial. Evaluate Answer: (1+2+3+4+5) 2 =225

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 a1.   2.  the recursion equation for an as a function of an-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 1, 4, 7, 10, 13, 16 d = 3 15, 10, 5, 0, -5, -10, d = -5 1, 1 2 ,0, d = -1/2

Examples Find the common difference for the arithmetic sequence whose formula is an = 6n + 3 Hint: Plug in 𝑎 1 & 𝑎 2 Answer: 6

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.

Examples Find the 10th term of the sequence 3, 5, 7, 9, ... n = 10;  a1 = 3, d = 2   The tenth term is 21.

Examples Find a formula for the sequence 1, 3, 5, 7, ... Hint: Work the sequence formula backwards 𝑎 𝑛 = 𝑎 1 + 𝑛−1 𝑑 𝑎 𝑛 =1+ 𝑛−1 2 𝑎 𝑛 =1+2𝑛−2  Answer

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.

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

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 5, 10, 20, 40, ... r = 2 -11, 22, -44, 88, ... r = -2 4, 8 3 , 16 9 , 32 27 , 64 81 𝑟= 2 3

Find the first 5 terms given the following Example Find the first 5 terms given the following 𝑎 1 =3, 𝑟=2 3, 6, 12, 24, 48 𝑎 1 =1, 𝑟=− 1 2 1, -0.5, 0.25, -0.125, 0.0625 1, − 1 2 , 1 4 , − 1 8 , 1 16

Any Term of a Geometric Sequence To find any term of a geometric sequence: 𝑎 𝑛 = 𝑎 1 • 𝑟 𝑛−1

Example Find the 12th term of the geometric sequence 5, 15, 45, …….. 𝑟= 15 5 =3 𝑎 12 =5 (3) 12−1 = 885,735

Example Find 𝑎 8 for the sequence: 0.5, 3.5, 24.5, 171.5 n = 8, r = 7, 𝑎 1 =0.5 𝑎 8 =0.5∗ 7 8−1 =411,771.5

Example The third term of a geometric sequence is 3 and the sixth term is 1/9.  Find the first term. Use 𝑎 3 as the first term. ___ , ___ , _3_ , ___ , ___ , _1/9_ Therefore, n = 4 for solving this problem.

Continued…. Now, work backward multiplying by 3 (or dividing by 1/3) to find the actual first term. a1 = 27

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

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. 𝑎 12 =2+ 12−1 2=24 𝑠 12 = 12(2+24) 2 =156

Example There are 2720 seats. 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 Series To find the sum of a certain number of terms of a geometric sequence: 𝑆 𝑛 = 𝑎 1 1− 𝑟 𝑛 1−𝑟 ; where 𝑟≠1 where Sn is the sum of n terms (nth partial sum),   a1 is the first term,  r is the common ratio

Example Evaluate using a formula: 𝑛=1 12 4 (0.3) 𝑛 =4 0.3 1− 0.3 12 1−0.3 =1.714

Example Evaluate the sum 𝑘=1 5 3 𝑘 =3 1− 3 5 1−3 =3 −242 −2 =363

Example Find the sum of the first 8 terms of the      sequence                      -5, 15, -45, 135, ... n = 8;  a1 = -5, r= -3 𝑆 8 =−5 1− −3 8 1− −3 = 32800 4 =8200

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….. Set up a model drawing for each "bounce".                 6.4, 5.12, ___, ___, ___  The common ratio is 0.8. 𝑎 𝑛 =6.4∗ 0.8 5−1 =2.62144 The ball will bounce approximately 2.6 feet