A sequenceis an ordered list of numbers, each one generated by a rule — 2, 5, 8, 11, … follows the rule “start at 2, add 3 each time.” A series is the sum of a sequence’s terms: 2 + 5 + 8 + 11 + …. Sequences are about the pattern; series are about the running total.
Arithmetic sequences — constant difference
In an arithmetic sequence, each term differs from the last by the same constant amount, called the common difference. The nth term is an = a1 + (n − 1)d, where a1is the first term and d is the common difference. This is exactly a linear function in disguise — the sequence 2, 5, 8, 11 is just y = 3x − 1, evaluated only at whole numbers.
Geometric sequences — constant ratio
In a geometric sequence, each term is the last one multiplied by a constant common ratio, r: 3, 6, 12, 24, … has r = 2. The nth term is an = a1·rn−1 — an exponential function in disguise, the same way the arithmetic sequence was a linear one.
Let me show you one — an infinite sum that isn’t infinite
Add 1 + 1/2 + 1/4 + 1/8 + … forever. Each term is half the one before — a geometric series with r = 1/2. Even though you’re adding infinitely many positive terms, the terms shrink fast enough that the running total never passes 2 — it creeps closer and closer without ever reaching it. The formula for an infinite geometric series, valid whenever |r| < 1, confirms it: S = a1/(1 − r) = 1/(1 − 0.5) = 2.
Why some infinite sums don’t settle down
If |r| ≥ 1 in a geometric series — or if a sequence doesn’t shrink toward zero at all — the running total either grows without bound or oscillates forever, and the infinite sum formula simply doesn’t apply. Checking |r| < 1 first is what tells you whether “adding forever” even has a sensible answer.
The mistake I see most
Mixing up the arithmetic and geometric formulas — using a common difference where a common ratio belongs. The fix is asking one question first: do consecutive terms differ by addition (arithmetic) or by multiplication (geometric)? That answer tells you exactly which formula applies before you write anything down.
Want to work through it together?
If the idea of summing infinitely many numbers still feels like it shouldn’t be allowed, that’s a completely reasonable reaction — and one worked example of a shrinking geometric series usually settles it for good.
