Arithmetic vs. Geometric Sequences and how to write their formulas

In this lesson, you will… Recognize patterns as sequences. Determine the next term in a sequence. Write explicit and recursive formulas for arithmetic and geometric sequencers. Use formulas to determine unknown terms of a sequence.

Arithmetic Sequence A sequence of terms in which the difference between any two consecutive terms is a constant. d = refers to the common difference (what’s being added or subtracted) Ex. 1, 4, 7, 10, 13… d = 3 Ex. 15, 10, 5, 0, -5, -10… d = -5

Geometric Sequence A sequence of terms in which the ratio between any two consecutive terms is a constant. r = refers to the common ratio (what’s being multiplied or divided) Ex. 5, 10, 20, 40… r = 2 Ex. -11, 22, -44, 88… r = -2

Identify the Sequences as Geometric or Arithmetic Arithmetic, d = 3 Arithmetic d = 1

Identify the Sequences as Geometric or Arithmetic Geometric, r = 3 Neither

Identify the Sequences as Geometric or Arithmetic Geometric r = .5 Arithmetic, d = 3

Finite Sequence A sequence that terminates (ends) Ex. Red, Orange, Yellow, Green, Blue, Indigo, Violet Ex. 1, 3, 5, 7, 9

Infinite Sequence A sequence that continues forever Ex. 1, 2, 3, 4, 5…

Determining infinite vs. finite If it includes a … and can go on forever, it is infinite. It is only finite if it is clearly defined. Finite arithmetic sequence: -2, -4, -6, -8 Infinite arithmetic sequence: -2, -4, -6, -8…

Writing formulas for Sequences An explicit formula for a sequence is a formula used for calculating each term of the sequence using the index (a term’s position in the sequence).

Writing Explicit Formulas For Arithmetic Sequences: an = a1 + d (n – 1) Where a1 is the first term, d is the common difference, and n is the nth term in the sequence. For Geometric Sequences: gn = g1 · rn-1 Where g1 is the first term, and r is the common ratio.

Practice Writing Formulas Use the explicit formula to determine the 5th term. an = a1 + d (n – 1) a5 = 9 + 3 (5 – 1) a5 = 9 + 3 (4) a5 = 9 + 12 a5 = 21

Practice Writing Formulas Use the explicit formula to determine the 5th term. gn = g1 · rn-1 g5 = g1 · 35-1 g5 = 3 · 34 g5 = 243

Practice Writing Formulas -5, -1, 3, 7, 11, 15, 19, 23 Type of Sequence? 50th Term using explicit formula? b) 1, 2, 4, 8, 16… c) 5, 3, 1, -1… Type of Sequence? Next term using explicit formula? Arithmetic, 191 Geometric, 5.62 x 10^14 Arithmetic, -3

Gauss’s Formula

In this lesson, you will… Compute a finite series. Use sigma notation to represent a sum of a finite series. Use Gauss’s method to compute finite arithmetic series. Write a function to represent the sum of a finite arithmetic series.

Series The sum of terms in a given sequence. The sum of the first n terms of a sequence is denoted by Sn. For example, S3 is the sum of the first three terms of a sequence.

Sigma Notation

Sigma Notation Continued Sum the values of a, starting at a1 and ending with an.

Using Sigma Notation for Finite Series A Finite Series is the sum of a finite number of terms. Example: 5 + 9 + 13 + 17 + 21

Using Sigma Notation for Finite Series Example: 3 + 6 + 12 + 24 + 48 + 96 + 192

Practice Rewrite the series as a sum.

Practice Rewrite each series using sigma notation.

Gauss’s Formula for finding the Sum of an Arithmetic Series Add the first term and the last term of the series, multiply the sum by the number of terms of the series, and divide by 2. N is the number of terms, a1 is the first term, an is the last term.

Gauss’s Formula Practice

Gauss’s Formula Practice 1 + 2 + 3 + … + 51 Identify the first term, last term, and number of terms. What’s the sum? 1326

Gauss’s Formula Practice 108 147 170 216

Challenge 1140

Euclid’s Method

In this lesson, you will… Generalize patterns to derive the formula for the sum of a finite geometric series Compute a finite geometric series

Geometric Series The sum of the terms of a geometric sequence

Sigma Notation Revisited

Euclid’s Method r represents what you’re multiplying by g1 is the first term n is number of terms gn is the last term

Euclid’s Method Example You won’t be given r. You have to find it.

Euclid’s Method Example

Euclid’s Method Practice 1 + 5 + 25 + 125 + 625 1 + (-2) + 4 + (-8) + 16 + (-32) D d r = 5, g1 = 1, gn = 625; 781 r = -2, g1 = 1, g5 = -32; -21 r = 2, g1 = 2^0 = 1, gn = 2^7 = 128; 255 r = (1/2), g1 = (1/2)^0 = 1, gn = (1/2)^8 = 1/256; 511/256 or 1.99609375

Infinite Geometric Series

In this lesson, you will… Write a formula for an infinite geometric series. Compute an infinite geometric series. Draw diagrams to model infinite geometric series. Determine whether series are convergent or divergent. Use a formula to compute a convergent infinite geometric series.

Divergent Series Infinite series Does not have a finite sum (sum is infinity) Common ratio (r) is greater than 1

Convergent Series Infinite series Finite sum (able to calculate) Common ratio (r) is between 0 and 1. The formula to compute a convergent geometric series S is:

Determining Convergent vs. Divergent The series is convergent because the common ratio is between 0 and 1.

Determining Convergent vs. Divergent

Determining Convergent vs. Divergent 1 1-3/4 =4

Determining Convergent vs. Divergent Convergent. Common ratio is between 0 and 1. ¼ 1-1/4 = 1/3

Determining Convergent vs. Divergent Ratio is greater than 1 (3/2) Infinity

Challenge Find the common ratio and compute the sum of the infinite geometric series. 8 + 2 + ½ + 1/8 … R = ¼ 8 1-1/4 = 32/3