Arithmetic progressions and the nth term
Algebra • Sequences
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Definition of an arithmetic progression (AP)
An arithmetic progression is a sequence in which each term after the first is produced by adding the same constant, called the common difference, to the previous term. Cause: constant addition between terms. Effect: linear change in term values and constant first differences. Limiting factors: the AP model applies only when consecutive differences are identical; variable or multiplicative changes do not qualify as arithmetic progressions.
Nth-term formula for linear sequences
The nth term of an arithmetic progression has the form a_n = a_1 + (n − 1)d, where a_1 is the first term and d is the common difference. Cause: adding d repeatedly to the first term (n−1) times. Effect: direct computation of any term without generating previous terms. Derivation method: observe the first few terms, calculate the constant difference d, substitute a_1 and d into the formula, and simplify. Limiting factors: indexing must match the chosen first-term position; if indexing starts at n = 0 use a_n = a_0 + nd.
Identifying arithmetic sequences from data
Identification requires checking consecutive differences. Cause: constant d yields identical first differences across the sequence. Effect: confirmation that the sequence is arithmetic and that the linear nth-term formula applies. Practical check: compute several first differences; identical results confirm an AP. Limiting factors: finite sample size can be misleading if differences coincide for a few terms but change later; check enough terms to be confident.
Quadratic sequences and second differences
Quadratic sequences produce a constant second difference when consecutive first differences are computed. Cause: an^2 + bn + c produces a linear first difference, whose own difference is constant. Effect: recognition that the nth term is quadratic in n and requires three parameters to specify. Method of deduction: use the constant second difference to determine 2a, then solve for b and c by substituting early terms into an^2 + bn + c. Limiting factors: only sequences with constant second differences are quadratic; higher-degree behaviour requires other methods.
Deduction methods for nth-term expressions
Linear nth terms deduce directly from first term and common difference. Cause: constant first difference simplifies to a_1 + (n−1)d. Effect: immediate formula for any n. Quadratic nth terms deduce by using second differences and solving a system for a, b, c or by fitting standard values for n = 1,2,3. Cause: constant second difference equals 2a, which yields a. Effect: substitution and simple algebra produce b and c. Limiting factors: arithmetic errors when solving simultaneous equations and inconsistent indexing produce incorrect coefficients.
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