Sequences
A sequence is an ordered list of numbers that follow a specific pattern or rule. Arithmetic sequences increase by a constant difference between consecutive terms, while geometric sequences multiply by a constant ratio. The sequence 2, 5, 8, 11 adds 3 each time, making it arithmetic with a common difference of 3.
Why it matters
Sequences appear throughout mathematics and real-world applications. In finance, compound interest follows geometric sequences where money grows by a constant percentage each year — $1000 at 5% annual interest becomes $1050, $1102.50, $1157.63. Population growth models use geometric sequences to predict changes over time. In physics, objects in free fall follow arithmetic sequences for velocity — dropping 9.8 m/s faster each second. Sequences form the foundation for calculus series and appear in computer algorithms. Architecture uses arithmetic sequences in step designs and geometric sequences in spiral structures. Understanding sequences is essential for CCSS.HSF.BF and CCSS.HSF.LE standards, preparing students for advanced topics like limits, derivatives, and mathematical modeling in college-level courses.
How to solve sequences
Sequences
- Arithmetic sequence: constant difference (d) between terms. aₙ = a₁ + (n−1)d.
- Geometric sequence: constant ratio (r) between terms. aₙ = a₁ × rn−1.
- To identify: check differences first, then ratios.
- Sum of arithmetic series: S = n/2 × (first + last).
Example: 2, 6, 18, 54: ratio = 3, geometric. a₅ = 2 × 3⁴ = 162.
Worked examples
Write the next 3 terms: 9, 19, 29, __, __, __
Answer: 39, 49, 59
- Find the common difference → d = 10 — 19 − 9 = 10. Each term increases by 10.
- Continue the pattern → 39, 49, 59 — 29 + 10 = 39, 39 + 10 = 49, 49 + 10 = 59.
Find the 12th term of: 7, 10, 13, 16, ...
Answer: 40
- Identify first term and common difference → a₁ = 7, d = 3 — First term is 7. Difference: 10 − 7 = 3.
- Use the nth term formula → aₙ = a₁ + (n − 1)d — The nth term of an arithmetic sequence is a₁ + (n − 1)d.
- Substitute → a_12 = 7 + (12 − 1) × 3 — Replace a₁ with 7, n with 12, d with 3.
- Calculate → 40 — 7 + 11 × 3 = 7 + 33 = 40.
Find the common difference and the 20th term: 2, 8, 14, 20, ...
Answer: d = 6, 20th term = 116
- Find the common difference → d = 8 − 2 = 6 — Subtract consecutive terms: 8 − 2 = 6.
- Use the nth term formula → a₂₀ = 2 + (20 − 1) × 6 — aₙ = a₁ + (n − 1)d with n = 20.
- Calculate → 116 — 2 + 19 × 6 = 2 + 114 = 116.
Common mistakes
- Confusing arithmetic and geometric patterns leads to errors like treating 2, 6, 18, 54 as arithmetic (difference of 4, 12, 36) instead of geometric (ratio of 3).
- Incorrect formula application produces wrong answers such as calculating the 8th term of 3, 7, 11, 15 as 3 + 8 × 4 = 35 instead of 3 + (8-1) × 4 = 31.
- Misidentifying the first term results in errors like using a₁ = 5 for the sequence 2, 5, 8, 11 instead of a₁ = 2, leading to incorrect calculations.