Mathematics
Arithmetic Sequences
An arithmetic sequence has a constant difference between consecutive terms (e.g. 3, 7, 11, 15, ... where d = 4). Two key formulas: the nth term a_n = a_1 + (n-1)·d, and the sum of the first n terms S_n = n/2·(a_1 + a_n).
How to Approach Arithmetic Sequences
Find the common difference d
Subtract any two consecutive terms. If the differences aren't constant, it's not arithmetic — it's something else (often geometric).
Apply the nth term formula
a_n = a_1 + (n-1)·d. Plug in the first term, common difference, and which term you want (e.g. the 50th).
Sum if asked
S_n = n/2·(a_1 + a_n). Or equivalently S_n = n/2·[2a_1 + (n-1)d]. Use whichever form matches what you've already computed.
Frequently Asked Questions
What's the difference vs geometric?+
Arithmetic has a constant difference (subtract consecutive terms). Geometric has a constant ratio (divide consecutive terms).
Can d be negative?+
Yes — the sequence then decreases. 100, 95, 90, ... has d = -5. The same formulas work; just plug in the negative value.
How is this used in real life?+
Simple interest, evenly spaced data points, depreciation models, fence-post problems. Anywhere something changes by a fixed amount each step.
Related Topics
More step-by-step guides in Mathematics and adjacent subjects.
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