Arithmetic Sequences

Pre-Calculus · Axiom Academy

One constant step at a time — how a single common difference builds a sequence, its formula, and its straight-line graph. An arithmetic sequence moves from each term to the next by adding the same number. Watch the sequence — every single jump is exactly +3 . That repeated jump is the common difference , written d . 2. To Reach Term n , Take n-1 Steps Here's the key insight: to get to the n th term you start at a_1 and take the step d a certain number of times. Reaching a_2 takes one step, a_3 takes two , and a_5 takes four — always one fewer step than the term number. Counting those steps gives the formula. a_1 = first term · d = common difference · n = term number Jump straight to the 100th term For we have a_1 = 2 and d = 3 . No need to list all hundred terms — the formula takes 99 steps at once: 3. Why the Graph Is a Straight Line Because the same amount d is added at every step, plotting the terms against their position gives points that all fall on one straight line. The slope of that line is exactly d : move one step right in n and the value rises by d . An arithmetic sequence is just a linear function of n . The sign of d tells you which way the line tilts — and therefore whether the sequence grows, holds steady, or shrinks: Line tilts up — the sequence is increasing. Flat line — every term is the same (a constant sequence). Line tilts down — the sequence is decreasing.

This is the written version of the interactive lesson above. See the full Pre-Calculus course.