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Arithmetic & Geometric Sequences: Find the nth Term

Arithmetic sequences add the same number each step (linear pattern). Geometric sequences multiply by the same number each step (exponential pattern). Two formulas, one strategy: identify, then plug in.

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Patterns are sequences

2, 5, 8, 11, ... — same amount added each step. 2, 6, 18, 54, ... — same multiplier each step. Both are sequences: ordered lists of numbers following a rule. Algebra 1 asks you to identify which type, find the rule, and use it to predict the 7th, 10th, or 100th term.

Arithmetic sequences (add)

an = a1 + (n − 1) · d a1 = first term,   d = common difference Subtract any term from the next to get d. Same throughout the sequence.

Example

Sequence: 5, 9, 13, 17, ...   What's the next term?

d = 9 − 5 = 4 a5 = 17 + 4 = 21
Practice

Next term

What is the next term in the arithmetic sequence: 5, 9, 13, 17, ...?

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Formula example: find the 8th term

a1 = 5,   d = 3 a8 = 5 + (8 − 1) · 3 = 5 + 21 = 26
Practice

nth term of an arithmetic sequence

Arithmetic sequence: a₁ = 5, d = 3. Find a₈.

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Geometric sequences (multiply)

an = a1 · r(n − 1) a1 = first term,   r = common ratio Divide any term by the previous to get r. Same throughout.

Example

Sequence: 5, 10, 20, 40, ...   What is the common ratio?

r = 10 / 5 = 2 Each term is double the previous.
Practice

Find the common ratio

Geometric sequence: 5, 10, 20, 40, ... What is the common ratio?

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Tell them apart

First check “subtract,” then “divide”

Compute a2 − a1 and a3 − a2. Same? → arithmetic. Different? Compute a2/a1 and a3/a2. Same? → geometric. Neither? → not a standard sequence.

Practice

Identify the sequence type

Is the sequence 2, 6, 18, 54, ... arithmetic or geometric?

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The big connection

An arithmetic sequence is a discrete linear function (the common difference is the slope). A geometric sequence is a discrete exponential function (the common ratio is the base b). Same math, different notation.

3-second recap

  • Arithmetic → add d each step. an = a1 + (n − 1) · d
  • Geometric → multiply by r each step. an = a1 · rn−1
  • Subtract for d. Divide for r. (n − 1), not n, in the formula.