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§ Expressions & Algebra

Sequences

§ Expressions & Algebra

Sequences

CCSS.HSF.BFCCSS.HSF.LE3 min read

Sequences appear everywhere from seating arrangements in theaters to payment schedules for loans. Students who master arithmetic and geometric sequences in CCSS.HSF.BF and CCSS.HSF.LE develop critical pattern recognition skills that transfer directly to advanced mathematics and real-world problem solving.

§ 01

Why it matters

Sequences form the foundation for understanding exponential growth in finance, population studies, and technology. When students analyze a loan with monthly payments of $450, $445, $440 (decreasing by $5 each month), they're working with arithmetic sequences. A bacteria culture that doubles every 3 hours follows a geometric sequence: 100, 200, 400, 800 bacteria. Engineers use sequences to design stadium seating where row 1 has 20 seats, row 2 has 24 seats, continuing with 4 additional seats per row. Investment portfolios growing at 7% annually follow geometric patterns. These mathematical structures appear in music (octaves doubling frequencies), architecture (spiral staircases with consistent step heights), and computer algorithms that process data in predictable patterns.

§ 02

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.

§ 03

Worked examples

Beginner§ 01

Write the next 3 terms: 7, 10, 13, __, __, __

Answer: 16, 19, 22

  1. Find the common difference → d = 3 — 10 − 7 = 3. Each term increases by 3.
  2. Continue the pattern → 16, 19, 22 — 13 + 3 = 16, 16 + 3 = 19, 19 + 3 = 22.
Easy§ 02

Find the 10th term of: 3, 5, 7, 9, ...

Answer: 21

  1. Identify first term and common difference → a₁ = 3, d = 2 — First term is 3. Difference: 5 − 3 = 2.
  2. Use the nth term formula → aₙ = a₁ + (n − 1)d — The nth term of an arithmetic sequence is a₁ + (n − 1)d.
  3. Substitute → a_10 = 3 + (10 − 1) × 2 — Replace a₁ with 3, n with 10, d with 2.
  4. Calculate → 21 — 3 + 9 × 2 = 3 + 18 = 21.
Medium§ 03

Find the common difference and the 20th term: 5, 11, 17, 23, ...

Answer: d = 6, 20th term = 119

  1. Find the common difference → d = 11 − 5 = 6 — Subtract consecutive terms: 11 − 5 = 6.
  2. Use the nth term formula → a₂₀ = 5 + (20 − 1) × 6 — aₙ = a₁ + (n − 1)d with n = 20.
  3. Calculate → 119 — 5 + 19 × 6 = 5 + 114 = 119.
§ 04

Common mistakes

  • Students confuse arithmetic and geometric sequences, writing 2, 6, 18, 54 as having a common difference of 4 instead of recognizing the common ratio of 3.
  • When finding the nth term, students forget to subtract 1 from n, calculating a₁₀ = 3 + 10 × 2 = 23 instead of a₁₀ = 3 + (10-1) × 2 = 21.
  • Students add the common ratio instead of multiplying, writing the next term of 4, 12, 36 as 36 + 3 = 39 instead of 36 × 3 = 108.
  • For series sums, students multiply by n instead of n/2, calculating S = 10 × (2 + 20) = 220 instead of S = 10/2 × (2 + 20) = 110.
§ 05

Frequently asked questions

How do I tell if a sequence is arithmetic or geometric?
Check differences first. If consecutive terms have the same difference (5, 8, 11: difference = 3), it's arithmetic. If differences vary but ratios are constant (3, 6, 12: ratio = 2), it's geometric. Calculate 2-3 pairs to confirm the pattern.
What's the fastest way to find the 50th term?
Use the formula rather than writing out terms. For arithmetic: aₙ = a₁ + (n-1)d. For geometric: aₙ = a₁ × r^(n-1). With sequence 7, 11, 15..., the 50th term is 7 + 49 × 4 = 203, much faster than listing all terms.
When do I use the sum formula versus finding individual terms?
Use the sum formula S = n/2 × (first + last) when problems ask for totals: "find the sum of the first 20 terms." Find individual terms when asked for specific positions: "what is the 15th term?" The context determines which approach to use.
Can a sequence be neither arithmetic nor geometric?
Yes. Fibonacci sequences (1, 1, 2, 3, 5, 8...) have neither constant differences nor ratios. Square numbers (1, 4, 9, 16...) form quadratic sequences. However, most CCSS problems focus on arithmetic and geometric sequences with their predictable formulas.
How do I check if my sequence formula is correct?
Substitute small values of n back into your formula. If aₙ = 3 + (n-1) × 4, then a₁ = 3, a₂ = 7, a₃ = 11. Compare these with the original sequence. If they match, your formula is correct.
§ 06

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