Arithmetic & Geometric Sequences Ellipsis Arithmetic sequence Geometric sequence

SEQUENCE Pattern involving an ordered arrangement of numbers, figures, or letters Terms Lower Bound Upper Bound 1, 3, 5, 7, 9, 11 Sequences can be with symbols or figures too!

ELLIPSIS … ELLIPSIS 1, 2, 3, 4, 5,… Sequences can be… FINITE or INFINITE Finite: sequence terminates Infinite: continues forever Use ellipsis to signify infinite sequence 1, 2, 3, 4, 5,…

Difference is usually referred to as ARITHMETIC SEQUENCE Difference is usually referred to as “Common Difference” (d) Sequence of terms, where the difference between consecutive terms stays constant Arithmetic Sequence Not Arithmetic Sequence +2 +3 +4 d = -5 d = undefined

ARITHMETIC SEQUENCE Arithmetic Not Arithmetic Arithmetic Arithmetic Determine whether the sequences below are arithmetic or not Arithmetic Not Arithmetic Arithmetic Arithmetic

GEOMETRIC SEQUENCE 1, 5, 25, 50, 250,… r = 2 r = undefined Sequence of terms, where the ratio between consecutive term stays constant Ratio is usually referred to as “common ratio” (r) Geometric Sequence Not Geometric Sequence 1, 5, 25, 50, 250,… ✕ 5 ✕ 5 ✕ 2 ✕ 5 ✕ 2 ✕ 2 ✕ 2 ✕ 2 ✕ 2 ✕ 2 r = 2 r = undefined

GEOMETRIC SEQUENCE Determine whether the sequences below are geometric or not r = 3 Geometric r = undefined 16, 8, 4, 1 Not Geometric r = -1/2 Geometric Geometric r = -2

Arithmetic sequence ____ Geometric sequence _____ Neither SEQUENCE PRACTICE Number of white triangles. Arithmetic sequence ____ Geometric sequence _____ Neither Finite Infinite 3 9 27 ✕3

Arithmetic sequence ____ Geometric sequence _____ Neither SEQUENCE PRACTICE Number of people. Arithmetic sequence ____ Geometric sequence _____ Neither Finite Infinite 1 2 4 8 ✕2

WORKSHEET TIME Time for some extra practice. Answer the four questions on the worksheet to the best of your ability. 8 min. time limit ANSWERS: Geometric sequence, *2, Finite Neither, since it isn’t a sequence, the last sentence of the directions doesn’t apply Geometric sequence, *2/3, Infinite Arithmetic sequence, +3, Finite

ARITHMETIC EXPLICIT an = a1 + d(n-1) 1, 3, 5, 7 1 3 5 7 Let’s take a simple arithmetic sequence for example: 1, 3, 5, 7 Now let’s label each term number: 1 3 5 7 1 2 3 4

ARITHMETIC EXPLICIT x y Let’s write out the sequence from the previous slide into a table for visual purposes: Therefore, we can rewrite the table into an equation: y = 2x – 1 What is the value of the 4th term in this sequence? y = 2(4) – 1 y = 8 – 1 y = 7 Term number x 1 2 3 4 y 5 7 Term value An explicit formula defines the value at a specific position in an arithmetic sequence.

ARITHMETIC RECURSIVE 1  3 x y Instead of using the relationship between the term number and term value, a recursive formula uses the relationship between a term value and the previous term’s value Term number x 1 2 3 4 y 5 7 Term value 1  3 Each term is the previous term PLUS the common difference! +2

RECURSIVE SEQUENCE What does this mean?

EXPLICIT OR RECURSIVE Explicit or Recursive? For this hour, Gerard is paid $10 more than what he was paid the previous hour. Gerard is paid $12. Answer: A = Recursive; B = Explicit I eat Cornflakes for breakfast. I place the toothpaste on my toothbrush before brushing my teeth. Answer: A = Explicit; B = Recursive

FORMULAS an = a1 + d(n-1) gn = g1 ✕ r(n-1) an = an-1 + d gn = gn-1 ✕ r Arithmetic Sequence Geometric Sequence Explicit Formula an = a1 + d(n-1) gn = g1 ✕ r(n-1) Recursive Formula an = an-1 + d gn = gn-1 ✕ r

GEOMETRIC FORMULAS There is a slight difference between arithmetic and geometric sequences. Instead or d, we have r. Explicit: gn = g1 ✕ r(n-1) The 3rd term is 3 multiplied by 32. 3 * 9 = 27 The 2nd term is 3 multiplied by 31. 3 * 3 = 9 Recursive: gn = gn-1 ✕ r The 3rd term is the 2nd term multiplied by the common ratio. 9 * 3 = 27 x 1 2 3 y 9 27 x 1 2 3 y 9 27