Sequences and Series

Grade 10 · Math · Study Guide

Arithmetic and geometric sequences, nth term formulas, and series sums.

A sequence is an ordered list of numbers where each number is called a term. Two types dominate high school math: arithmetic and geometric. In an arithmetic sequence, consecutive terms differ by a constant amount called the common difference (d). In a geometric sequence, each term is multiplied by a constant ratio called the common ratio (r).

For an arithmetic sequence with first term a₁ and common difference d, the nth term is: aₙ = a₁ + (n-1)d. The sequence 3, 7, 11, 15, ... has a₁ = 3 and d = 4. The 10th term is a₁₀ = 3 + (10-1)(4) = 3 + 36 = 39.

For a geometric sequence with first term a₁ and common ratio r, the nth term is: aₙ = a₁ · rⁿ⁻¹. The sequence 2, 6, 18, 54, ... has a₁ = 2 and r = 3. The 5th term is a₅ = 2 · 3⁴ = 162.

A series is the sum of the terms of a sequence. For an arithmetic series, the sum of the first n terms is: Sₙ = n/2 · (a₁ + aₙ) or equivalently Sₙ = n/2 · (2a₁ + (n-1)d). For a finite geometric series: Sₙ = a₁ · (1 - rⁿ)/(1 - r), when r ≠ 1. The story of the mathematician Gauss summing 1 + 2 + ... + 100 = 5050 in seconds as a child is essentially discovering the arithmetic series formula.

Key terms

Arithmetic Sequence

Each term increases by a constant d (common difference): aₙ = a₁ + (n-1)d.

Geometric Sequence

Each term is multiplied by a constant r (common ratio): aₙ = a₁ · rⁿ⁻¹.

Common Difference (d)

The fixed amount added between consecutive terms in an arithmetic sequence.

Common Ratio (r)

The fixed multiplier between consecutive terms in a geometric sequence.

Series

The sum of the terms of a sequence; Sₙ denotes the sum of the first n terms.

Flashcards

Arithmetic sequence nth term

aₙ = a₁ + (n-1)d

Geometric sequence nth term

aₙ = a₁ · rⁿ⁻¹

Common difference d

aₙ - aₙ₋₁; the constant added in an arithmetic sequence

Common ratio r

aₙ / aₙ₋₁; the constant multiplied in a geometric sequence

Arithmetic series sum Sₙ

Sₙ = n/2 · (a₁ + aₙ)

Geometric series sum Sₙ

Sₙ = a₁ · (1 - rⁿ)/(1 - r), r ≠ 1

Practice questions

1. In the arithmetic sequence 5, 9, 13, 17, ..., what is the common difference?

   1. 5
   2. 4
   3. 3
   4. 9
   Show answer

   Answer: 4

   d = 9 - 5 = 4 (or 13 - 9 = 4). The common difference is 4.

2. For the sequence 3, 6, 12, 24, ..., the common ratio r = ___.

   Show answer

   Answer: 2

   r = 6/3 = 2 (or 12/6 = 2). Each term is multiplied by 2.

3. The formula for the nth term of an arithmetic sequence is aₙ = a₁ + (n-1)d.

   Show answer

   Answer: True

   This is the standard formula: start at a₁ and add d a total of (n-1) times to reach the nth term.

4. What is the 6th term of the arithmetic sequence 2, 5, 8, 11, ...?

   1. 14
   2. 17
   3. 20
   4. 23
   Show answer

   Answer: 17

   a₁=2, d=3. a₆ = 2 + (6-1)·3 = 2 + 15 = 17.

5. A series is the product of the terms of a sequence.

   Show answer

   Answer: False

   A series is the SUM of the terms of a sequence, not the product.

Fun facts

• The legend goes that young Carl Friedrich Gauss solved the sum 1+2+...+100 in seconds by pairing numbers (1+100, 2+99, ... = 50 pairs of 101), discovering Sₙ = n(n+1)/2 on his own.
• Geometric sequences appear in music: each octave doubles the frequency of a note, and the equal-tempered scale divides each octave into 12 steps using the ratio 2^(1/12).

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