4.1 Numerical and Algebraic Patterns, Sequences & Formulas
Key Takeaways
- An arithmetic sequence adds a constant common difference d, with explicit formula aₙ = a₁ + (n − 1)d.
- A geometric sequence multiplies by a constant common ratio r, with explicit formula aₙ = a₁ · r^(n−1).
- Constant first differences in a table signal a linear rule; constant second differences signal a quadratic rule; a constant ratio signals an exponential rule.
- The exponent in aₙ = a₁ · r^(n−1) is n − 1, not n, because the first term has already been counted before any multiplication occurs.
- Figural patterns are generalized by counting what stays fixed (the constant) and what repeats per step (the coefficient), giving a rule of the form output = (rate)(step) + (constant).
4.1 Numerical and Algebraic Patterns, Sequences & Formulas
Competency 2 opens with pattern generalization because it is the conceptual entry point to all of algebra: a pattern is a function waiting to be written symbolically. Skill 1 names four representations — tables, graphs, written descriptions, and formulas — and items typically hand you one and ask for another.
Arithmetic sequences
An arithmetic sequence adds a fixed common difference d between consecutive terms.
- Recursive form: aₙ = aₙ₋₁ + d, with a₁ given
- Explicit form: aₙ = a₁ + (n − 1)d
The recursive form tells you how to get the next term; the explicit form tells you the n-th term directly. Items that ask for the 40th term are testing whether you reach for the explicit form instead of listing.
Sequence: 7, 11, 15, 19, ... Find the 30th term. a₁ = 7, d = 4. So a₃₀ = 7 + (30 − 1)(4) = 7 + 116 = 123.
The single most common error is using n rather than n − 1. Check it against a known term: a₁ should return 7, and 7 + (1 − 1)(4) = 7 confirms the formula. Using 7 + 30(4) = 127 gives the 31st term.
An arithmetic sequence graphs as collinear points, and its explicit rule is linear in n, with d playing the role of slope.
Geometric sequences
A geometric sequence multiplies by a fixed common ratio r.
- Recursive form: aₙ = r · aₙ₋₁
- Explicit form: aₙ = a₁ · r^(n−1)
Sequence: 5, 15, 45, 135, ... Find the 7th term. a₁ = 5, r = 3. So a₇ = 5 · 3⁶ = 5 · 729 = 3,645.
Ratios may be fractional (decay) or negative (alternating signs). The sequence 64, −32, 16, −8, ... has r = −1/2, and a₆ = 64(−1/2)⁵ = 64(−1/32) = −2.
To find r from any two consecutive terms, divide: r = aₙ / aₙ₋₁. To find d, subtract.
Reading a rule from a table
+---------------------------------------------------------------------------+
| Finite Difference Diagnostic |
+---------------------------------------------------------------------------+
| First differences CONSTANT -> LINEAR y = mx + b |
| Second differences CONSTANT -> QUADRATIC y = ax^2 + bx + c |
| Ratios of consecutive y CONSTANT -> EXPONENTIAL y = a * b^x |
+---------------------------------------------------------------------------+
Linear example. x: 1, 2, 3, 4 and y: 5, 9, 13, 17. First differences are 4, 4, 4 — constant. So y = 4x + b; substituting (1, 5) gives 5 = 4 + b and b = 1, so y = 4x + 1.
Quadratic example. x: 1, 2, 3, 4, 5 and y: 3, 6, 11, 18, 27. First differences: 3, 5, 7, 9 — not constant. Second differences: 2, 2, 2 — constant. The leading coefficient is (second difference)/2 = 1, so the rule is x² plus a linear correction. Testing y = x² + 2 gives 3, 6, 11, 18, 27. Correct: y = x² + 2.
Exponential example. x: 0, 1, 2, 3 and y: 6, 18, 54, 162. Differences 12, 36, 108 grow, but ratios 18/6 = 3, 54/18 = 3, 162/54 = 3 are constant. So y = 6 · 3ˣ.
That the leading coefficient of a quadratic equals half the constant second difference is a genuinely useful shortcut: if second differences are 6, the x² coefficient is 3.
Generalizing figural patterns
Figural items show three or four stages of a growing figure and ask for stage n or for the stage at which some total is reached. The reliable method is to separate what stays the same from what is added each step.
A pattern of square tiles: Stage 1 has 5 tiles, Stage 2 has 8, Stage 3 has 11, Stage 4 has 14. Each stage adds 3 tiles, so the rate is 3. Working backward to a "Stage 0" gives 5 − 3 = 2, the constant. Rule: tiles = 3n + 2. Check: 3(4) + 2 = 14. ✓
Using this to answer backward questions is standard: at what stage does the figure have 62 tiles? Solve 3n + 2 = 62, giving n = 20.
Perimeter patterns often behave differently from area patterns in the same figure. A row of n unit squares has area n but perimeter 2n + 2, because two vertical edges persist no matter how long the row grows. Items pair these to test whether candidates model the actual structure rather than assuming every pattern is proportional.
Translating written descriptions
The last representation is verbal, and precision about word order matters.
| Description | Expression |
|---|---|
| Five more than twice a number | 2n + 5 |
| Five less than twice a number | 2n − 5 |
| Twice the sum of a number and five | 2(n + 5) |
| The quotient of a number and four, decreased by seven | n/4 − 7 |
| Seven decreased by the quotient of a number and four | 7 − n/4 |
The pairs differ only in order, and both members of each pair appear as answer options. "Less than" and "subtracted from" reverse the reading order; "decreased by" does not.
A table shows x = 1, 2, 3, 4, 5 with y = 4, 9, 18, 31, 48. What type of function models the data?
In the geometric sequence 800, 200, 50, 12.5, ..., what is the sixth term?
A tile pattern has 7 tiles at Stage 1, 12 at Stage 2, and 17 at Stage 3. At which stage will the figure contain 92 tiles?