Sequences and Series

Grade 9 · Math · Study Guide

Identify arithmetic and geometric sequences, write explicit and recursive formulas, and calculate series sums.

A sequence is an ordered list of numbers called terms. An arithmetic sequence has a constant difference (common difference, d) between consecutive terms. For example, 3, 7, 11, 15, ... has d = 4. The explicit formula is aₙ = a₁ + (n − 1)d, and the recursive formula is aₙ = aₙ₋₁ + d.

A geometric sequence has a constant ratio (common ratio, r) between consecutive terms. For example, 2, 6, 18, 54, ... has r = 3. The explicit formula is aₙ = a₁ · r^(n−1), and the recursive formula is aₙ = r · aₙ₋₁.

A series is the sum of the terms of a sequence. The sum of the first n terms of an arithmetic series is Sₙ = n/2 · (a₁ + aₙ) or equivalently Sₙ = n/2 · [2a₁ + (n − 1)d]. The sum of the first n terms of a geometric series is Sₙ = a₁(1 − rⁿ)/(1 − r) when r ≠ 1.

To identify the type of sequence, check whether there is a common difference (arithmetic) or a common ratio (geometric). Neither means it is neither type — for example, 1, 4, 9, 16, ... (perfect squares) is neither arithmetic nor geometric.

Key terms

Arithmetic sequence
A sequence where each term is obtained by adding a fixed common difference d to the previous term
Geometric sequence
A sequence where each term is obtained by multiplying the previous term by a fixed common ratio r
Explicit formula
A formula that expresses the nth term of a sequence directly in terms of n
Recursive formula
A formula that defines each term of a sequence using one or more previous terms
Arithmetic series
The sum of the terms of an arithmetic sequence
Geometric series
The sum of the terms of a geometric sequence

Flashcards

Arithmetic explicit formula
aₙ = a₁ + (n − 1)d
Geometric explicit formula
aₙ = a₁ · r^(n−1)
Sum of arithmetic series
Sₙ = n/2 · (a₁ + aₙ) or Sₙ = n/2 · [2a₁ + (n−1)d]
Sum of geometric series
Sₙ = a₁(1 − rⁿ) / (1 − r), for r ≠ 1
Find common difference
d = a₂ − a₁ (subtract any term from the next)
Find common ratio
r = a₂/a₁ (divide any term by the previous one)
10th term of 1, 3, 5, 7, ...
a₁₀ = 1 + (10−1)·2 = 1 + 18 = 19
4th term of 5, 10, 20, 40, ...
a₄ = 5 · 2³ = 5 · 8 = 40

Questions

  1. Find the 10th term of the arithmetic sequence: 5, 9, 13, 17, ...

    1. 37
    2. 41
    3. 45
    4. 49
    Show answer

    Answer: 41

    Common difference d = 4. a₁₀ = 5 + (10 − 1) · 4 = 5 + 36 = 41.

  2. In the geometric sequence 4, 12, 36, 108, ..., the common ratio is __.

    Show answer

    Answer: 3

    r = 12/4 = 3. Verify: 12·3 = 36, 36·3 = 108. Consistent.

  3. What is the sum of the first 6 terms of the arithmetic series: 2 + 5 + 8 + 11 + ...

    1. 51
    2. 57
    3. 60
    4. 54
    Show answer

    Answer: 57

    d = 3, a₁ = 2. The 6th term: a₆ = 2 + 5·3 = 17. S₆ = 6/2 · (2 + 17) = 3 · 19 = 57.

  4. The sequence 2, 6, 18, 54, 162 is an arithmetic sequence.

    Show answer

    Answer: False

    An arithmetic sequence has a common difference; 6/2 = 3, 18/6 = 3, so this is a geometric sequence with r = 3, not arithmetic.

  5. Find the 5th term of the geometric sequence where a₁ = 3 and r = 2.

    1. 24
    2. 48
    3. 32
    4. 96
    Show answer

    Answer: 48

    a₅ = 3 · 2^(5−1) = 3 · 2⁴ = 3 · 16 = 48.

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