Number Patterns

Identify and work with linear and quadratic sequences

Linear Number Patterns

A linear number pattern has a constant (common) difference between consecutive terms. The general term is Tn = a + (n − 1)d, where a is the first term, d is the common difference, and n is the term number.

Example

Finding the General Term

Pattern: 3, 7, 11, 15, 19, ... Common difference d = 7 − 3 = 4 First term a = 3 Tn = 3 + (n − 1)(4) = 3 + 4n − 4 = 4n − 1 Check: T1 = 4(1) − 1 = 3 ✓, T2 = 4(2) − 1 = 7 ✓

Quadratic Number Patterns

If the first differences are not constant but the second differences are, the pattern is quadratic. The general term has the form Tn = an² + bn + c. Use the second difference to find 'a': second difference = 2a.

Example

Quadratic Pattern Example

Pattern: 2, 6, 12, 20, 30, ... First differences: 4, 6, 8, 10 Second differences: 2, 2, 2 (constant → quadratic) 2a = 2, so a = 1 Tn = n² + bn + c Using T1 = 2: 1 + b + c = 2 Using T2 = 6: 4 + 2b + c = 6 Solving: b = 1, c = 0 Tn = n² + n = n(n + 1)

Note

Remember

Linear: constant first difference → Tn = a + (n − 1)d. Quadratic: constant second difference → Tn = an² + bn + c. Always verify by substituting n = 1, 2, 3 to check your formula.

Key Vocabulary

SequenceAn ordered list of numbers following a rule

TermEach number in a sequence

Common differenceThe constant difference between consecutive terms in a linear pattern

General termA formula (Tn) that gives any term in the sequence

QuadraticA pattern whose general term involves n²

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