Math & Statistics · 等差数列（Arithmetic Sequence） · Updated 2026-10-25
What Is Arithmetic Sequence?
Definition
The nth term is an = a1+(n−1)d with common difference d, and the sum is Sn = n(a1+an)/2 — the essence being pairing from both ends: in 1+2+…+100, pairs (1,100), (2,99)… each sum to 101 with 50 pairs, giving 5050 (Gauss's trick). Arithmetic structure appears in equal-principal mortgages (fixed monthly decrement) and straight-line depreciation.
Formula
an = a1 + (n−1)d; Sn = n(a1+an)/2
Worked Example
Equal-principal loan of 1M over 30 years: first payment 1M/360 + 1M×0.3% = 5,806, decreasing by 1M/360×0.3% ≈ 8.3 yuan monthly — a textbook arithmetic sequence.
Common Misconception
Applying the arithmetic sum to geometric contexts (or vice versa) — arithmetic is additive structure (constant difference), geometric is multiplicative (constant ratio). Ask whether each step "adds a fixed amount" or "multiplies by a fixed factor."
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