Figurate and quadratic sequences explained

Algebra • Sequences

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Find a for the sequence 2, 6, 12, 20,...

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Second difference = 2 so a = 1 because a = second difference / 2.

Key concepts

What you'll likely be quizzed about

Figurate numbers: definition and scope

Figurate numbers represent counts of equally spaced points arranged to form geometric shapes. Triangular, square and cube sequences provide the most common figurate examples. The concept limits to integer index n ≥ 1 and to exact formulas for each shape.

Triangular numbers

Triangular numbers count points that form an equilateral triangle. The nth triangular number equals n(n + 1)/2. The sequence starts 1, 3, 6, 10, 15, ... because successive addition of natural numbers (1, 2, 3, 4, ...) produces successive triangular values.

Square and cube numbers

Square numbers equal n^2 and list as 1, 4, 9, 16, 25, ... because each term multiplies an integer by itself. Cube numbers equal n^3 and list as 1, 8, 27, 64, ... because each term multiplies an integer by itself twice more. Both sequences require integer n ≥ 1 and produce constant polynomial forms of degree 2 and degree 3 respectively.

Quadratic sequences and second differences

Quadratic sequences follow a general nth-term of the form an^2 + bn + c where a, b and c are constants and a ≠ 0. The second differences between successive terms remain constant because the difference of a quadratic is linear and the difference of a linear sequence is constant. Constant second difference provides a direct diagnostic for quadratic behaviour.

Finding the nth term from differences

The constant second difference equals 2a, so a equals half the second difference. The first term and first difference supply linear equations to solve for b and c. Solving a small system of linear equations or using formulae derived from initial terms produces the coefficients a, b and c. The method requires at least three consecutive terms of the sequence.

Recognition and limitations

Recognition uses the pattern of differences: equal first differences indicate an arithmetic sequence; equal second differences indicate a quadratic sequence. The approach fails if given insufficient consecutive terms or if terms follow a different polynomial degree or non-polynomial rule. Verification by substituting n into the derived formula confirms correctness.

Key notes

Important points to keep in mind

Triangular numbers use formula n(n + 1)/2 for n ≥ 1.

Square numbers equal n^2 and cube numbers equal n^3 with integer n ≥ 1.

Quadratic sequences show constant second differences; second difference = 2a.

Solve three simultaneous equations from first three terms to obtain a, b and c if necessary.

Verify the derived nth term by substituting several n values from the original sequence.

Constant first differences indicate arithmetic sequences, not quadratic sequences.

Second-difference method assumes polynomial rules; check for non-polynomial patterns over more terms.

Always state domain restriction n ∈ ℕ (n ≥ 1) when giving an nth-term formula.