Arithmetic Progressions — Fundamental Concepts

Fundamental Concepts

Symbols and formulas

a: first term; d: next term minus current term.
n: positive integer position or number of terms, as specified.
a_n = a + (n − 1)d: one term.
S_n = n(a + l)/2: total when both endpoints are known.
S_n = n[2a + (n − 1)d]/2: total when a and d are known.
a_m − a_k = (m − k)d: recover the step from two positions.

Decision sequence

First establish that the sequence has a constant difference. Identify the measured quantity: a row's seats, one deposit or a cumulative balance. Then decide whether the question asks for a single term, a total, membership or a count. Those are different operations even when they use the same data.

Boundary checks

A position must be a positive integer. A finite sequence also imposes an upper limit. A zero common difference requires direct handling because division by d is unavailable. Negative differences are valid; preserve their sign.

For multiples in an interval, find the first and last permitted multiples before counting. Use n = (last − first)/d + 1 when d is positive and endpoints are included. Do not confuse equal percentage growth with equal additions. Finally, substitute a recovered first term and difference into the original conditions to check the result independently.

Check the first and second terms before trusting an indexed formula; this catches an extra common-difference step.