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Sequences & Series

Understand arithmetic and geometric sequences with explicit and recursive formulas. Learn to find terms, common differences, ratios, and solve sequence problems with examples.

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Sequences & Series

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  1. Sequences & Series Explicit & Recursive Formulas

  2. Arithmetic & Geometric Sequences • A Sequence is a list of things (usually numbers) that are in order. • 2 Types of formulas: • Explicit & Recursive Formulas

  3. Arithmetic sequences • A list of numbers that related to each other by a rule. • The terms are the numbers that form the sequence. • Goes from one term to the next by adding and subtracting. • Two types of sequences with two types of formulas. 1. Arithmetic: Explicit & Recursive 2. Geometric: Explicit & Recursive

  4. Arithmetic Sequences • A sequence in which the common difference between terms is always the same number. • ADD to get to the next term. • 2 types of formula to find either the NEXT number or A NUMBER in the sequence. Explicit vs Recursive

  5. RECURSIVE FORM • Finds the NEXT Term in the sequence an= an-1 + d Common Difference Previous Term • the nth term (term you want) • n = the term number • means the number before nth term • d = common difference (can be negative)

  6. Examples – Finding the Common Difference (d) • Find the common difference. • 3, 5, 7, 9, 11 • -2, -4, -6, -8, -10 • -3, 0, 3, 6, 9 • 19, 10, 1, -8, -17

  7. Recursive Form is… NEXT Number = CURRENT Term + d an = an-1 + d

  8. Recursive Form Examples 1) Find the 4th term given: a4 = ? , a3 = 6 , d = 2 2) Find the 3rd term given: a3 = ? , a2 = 17 , d = -7 3) Find the 6th term given: a6 = ? , a5 = 2 , d = -5

  9. Write the recursive form for each sequence. Then find the next three terms. 4) 12, 9, 6, 3, 0, … 5) 56, 61, 66, 71, … 6) -14, -24, -34, -44, …

  10. Explicit Formula One less term than the term number • Finds any number in the sequence. • the nth term (term you want) • n = the term number • d = common difference (can be negative) • ) Common Difference First Term

  11. Example #1 Find the 10th term of our sequence: 4, 6, 8, 10… 1. Identify all your terms. n = _____ = _____ d = _____ 2. Substitute values into our formula a a a 22

  12. Example #2 Find the 25th term of our sequence: 6, 10, 14, 18, … 1. Identify all your terms. n = _____ a = _____ d = _____ 2. Substitute values into our formula a a a

  13. You try • Consider the sequence {6, 17, 28, 39, 50, …} • Find the 12th term. • Find the 50th term. • Find the 100th term.

  14. Application • A bag of dog food weighs 8 pounds at the beginning of day 1. Each day, the dogs are feed 0.1 pound of food. How much does the bag of dog food weigh at the beginning of day 30?

  15. Steps 1. Determine if the sequence is Arithmetic -Do you add or subtract by the same amount from one term to the next? 2. Ask if you want to find the next term or a term in the sequence? 3. Find the common difference. -The number you add or subtract 4. Create a Recursive or Explicit formula -State first term or previous term,

  16. Geometric Sequences • A sequence in which the ratio between consecutive terms is always the same number. • A geometric sequence is formed by multiplying a term in the sequence by a fixed number to find the next term. • Common Ratio- the fixed number that you multiply each term in the sequence.

  17. Geometric Sequences • A sequence in which the common ratio between terms is always the same number. • MULTIPLY to get to the next term. • 2 types of formula to find either the NEXT number or A NUMBER in the sequence. Explicit vs Recursive

  18. RECURSIVE FORM • Finds the NEXT Term in the sequence an= an-1∙ r Common Ratio Previous Term • the nth term (term you want) • n = the term number • means the number before nth term • r = common ratio (can be a fraction)

  19. Find the common ratio of each sequence. • 750, 150, 30, 6,… • 2. 9, -36, 144, -576,… • 8, -24, 72, -216,…

  20. Write the Recursive for each problem. • 750, 150, 30, 6,… • -3, -6, -12, -24,… • 9, -36, 144, -576,…

  21. Explicit Formula One less term than the term number • Finds any number in the sequence. • the nth term (term you want) • n = the term number • r = common ratio (can be a fraction) Common Ratio First Term

  22. Find a10, a33 , a5 • 750, 150, 30, 6,… • -3, -6, -12, -24,… • 9, -36, 144, -576,… • 8, -24, 72, -216,…

  23. Steps 1. Determine if the sequence is Geometric -Do you multiply or divide by the same amount from one term to the next? 2. Ask if you want to find the next term or a term in the sequence? 3. Find the common ratio. -The number you multiply or divide 4. Create a Recursive or Explicit formula -State first term or previous term,

  24. Graphs • Arithmetic Graphs: LINEAR • Geometric Graphs: EXPONENTIAL

  25. Arithmetic mean • Of any two number is the average of the two numbers. • Use the arithmetic mean to find a missing term of an arithmetic sequence. • Two terms of an arithmetic sequence and their arithmetic mean lie on the same line.

  26. Geometric sequence • Of any two positive numbers by taking the positive square root of the product of the two numbers. • Use the geometric mean to find a missing term of a geometric sequence.

  27. Arithmetic & Geometric Series • A indicated Sum of terms of a sequence is called a Series • OR…. • ASumof an infinite sequence it is called a "Series" • (it sounds like another name for sequence, but it is actually a sum).

  28. Example

  29. Arithmetic Series Number of terms First term nth term Sum of Numbers in Sequence Must Know first term, last term, and number of terms • is the first term • n is the number of terms

  30. Example 1: • Find the sum of the first 50 positive integers.

  31. Examples: 1. Write the related series for each finite sequence then evaluate. 21, 18, 15, 12, 9, 6, 3 Solution: • Evaluate the series: (8 terms) 5, 13, 21…61 Remember: These are the SUM of the numbers.

  32. Geometric Series is the first term r is the "common ratio" between terms n is the number of terms • Need to know three things:  the first term, how many terms to add and the common ratio!!

  33. Example 2: • Find the sum of the first six terms of a geometric series for which

  34. Examples: • Evaluate series to the given term. • Evaluate series. …, n = 6 *Don’t forget we are adding the terms not multiply by .5 each term.

  35. Example 3:

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