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8.2 – Arithmetic Sequences

Objectives

  • Define and identify an arithmetic sequence.
  • Use the simple, explicit, or recursive formulas to solve problems about arithmetic sequences.
  • Find the missing term in an arithmetic sequence.
  • Find the common difference between terms of an arithmetic sequence.
  • Identify the graph that represents a given arithmetic sequence.

 

Key Terms

  • Arithmetic Sequence – A sequence of numbers in which the difference between any two consecutive terms is the same. The first term of the sequence is often denoted by a1 and the common difference by d.
  • Common Difference – The constant value between any two consecutive numbers in an arithmetic sequence.
  • Consecutive Numbers – Pairs of numbers that appear next to each other in a sequence.
  • Explicit Formula – A formula that defines the nth term of a sequence using only the index, n.
  • Recursive Formula – A rule that defines the nth term of a sequence or series in terms of one or more previous terms.
  • Term – A number, or item, that represents a specific position in a sequence.
  • Value – The actual quantity of a given term in a sequence of numbers.

 

Notes

  • The difference between any two consecutive numbers in an arithmetic sequence is a constant value.
  • When the difference between the terms in a sequence is a constant number, the sequence is an arithmetic sequence.

 

  • Steps for finding the missing number in an arithmetic sequence

Alg1A 08.02 Steps

 

  • To find the nth term in the sequence, you can use the simple formula.
    • Start with the common difference between each term.
    • Make the common difference the coefficient of n — this is the first part of your rule.
    • Subtract the common difference from the first value of the arithmetic sequence to get the second part of your rule.
    • Add the two parts together to complete the rule.

 

  • First, learn the parts of the sequence: terms and values

Alg1A 08.02 Terms Values

 


  • To find the 100th term in the following example, there are two ways

Alg1A 08.02 The Nth term

 

The rule works like a function. Substitute any term number for n and then simplify. If the rule is correct, the result should be one of the values of the arithmetic sequence:

F(n) = 5n – 1
F(1) = 5(1) – 1 = 5 – 1 = 4
F(2) = 5(2) – 1 = 10 – 1 = 9
F(3) = 5(3) – 1 = 15 – 1 = 14
F(4) = 5(4) – 1 = 20 – 1 = 19

The rule works! You can now quickly find the 100th term in the arithmetic sequence — just find F(100).

 


  • Explicit Formula
    • The text says that in the explicit formula, n = any term in the sequence.
    • This does not mean that nis a number in the sequence.
    • It means n = the position of any number in the sequence.
      • Ex. In 3, 11, 19, 27, 35, …  if n = 2, the number in the sequence is 11 (the second number in the sequence)!

Alg1A 08.02 Explicit Formula

Alg1A 08.02 Explicit Formula

 

  • Example

Alg1A 08.02 Explicit Formula Ex

 


  • Recursive FormulaAlg1A 08.02 Recursive Formula

Alg1A 08.02 Recursive Formula 2

Alg1A 08.02 Recursive Formula 3

 


 

Examples from Quiz

  • Find the common difference between the terms of the following arithmetic sequence.
    • -6, -2, 2, 6, 10, …
    • Answer: 4
  • Which of the following choices is the common difference between the terms of this arithmetic sequence?
    • 55, 50, 45, 40, 35, …
    • Answer: -5
  • Table of the Arithmetic Sequence: F(x) = 4x – 6.
    • Row 1 – x: 1, 2, 3, 4…
    • Row 2 – F(x): -2, 2, 6, 10
  • Find the common difference between the terms of this arithmetic sequence:
    • 3x + 9y, 6x + 5y, 9x + y, 12x – 3y, 15x – 7y, …
    • Answer: 3x-4y
  • Denver, the capital of Colorado, is not called the “Mile-High City” for nothing. One of the steps in the front stairway of the state capitol building is marked as being exactly one mile above sea level.
    • One mile = 63,360 inches, and the risers of the steps of the capitol building are 7 inches high. Heather is visiting the capitol, and as she climbs up from the one-mile step, her altitude above sea level, in inches, is given by the following arithmetic sequence:
    • 63,360, 63,367, 63,374, 63,381, 63,388, …
  • The following sequence is an arithmetic sequence. You are not shown very many terms of this sequence, but you are shown enough to fill in the missing number. What number goes in the blank?
    • 7, , 19, …
    • Answer: 13

 

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