Arithmetic, Geometric and Mixed Series

Once you can read differences, you need to recognise the three engines that drive most series: adding a constant, multiplying by a constant, and a mixed rule that combines the two. Naming the engine quickly is what turns a 40-second struggle into a 15-second answer, so this topic is pure pattern vocabulary. Learn to name the engine on sight and the rest of the question — computing the next term — is almost always trivial arithmetic you can finish in your head.

Arithmetic Series

A fixed number is added or subtracted each step, so the first differences are constant. These are the gentlest series and the easiest to verify — once two gaps match, you can trust the rule. Watch for a subtractive version where the terms fall by a constant, which candidates sometimes misread as an error in the sequence.

Arithmetic: next term = previous term + d (d is constant)

Geometric Series

Each term is multiplied or divided by a constant ratio, so terms grow or shrink fast. If you see numbers leap upward, test the ratio immediately. A common bank twist uses a fractional ratio, such as ÷2 each step, producing a shrinking series that still follows a clean geometric rule.

Geometric: next term = previous term × r (r is constant)

Mixed and Two-Step Series

The hardest common type applies a changing operation: ×2 then +1, repeated, or ×2, ×3, ×4 with a rising multiplier. Test whether the operation itself follows a pattern rather than staying fixed. Another mixed form multiplies and then adds a constant each step, which you detect by checking whether (term − small constant) gives a clean ratio. When a single operation clearly fails, a two-step rule is almost always the answer, so reach for it early rather than as a last resort.

Solved Examples

Example 1: 5, 8, 11, 14, ? is arithmetic with d = 3, giving 17.

Example 2: 3, 6, 12, 24, ? is geometric with r = 2, giving 48.

Example 3: 2, 5, 11, 23, ? follows ×2 then +1: 2×2+1 = 5, 5×2+1 = 11, 23×2+1 = 47.

✗ Reading 2, 5, 11, 23 as random  |  ✓ Testing ×2 + 1, which fits every step

Verifying a Rule Before You Commit

Speed is worthless without a check, so confirm any rule on at least two consecutive steps before you trust it. A rule that fits the jump from the first to the second term but fails from the second to the third was never the real rule — it was a coincidence. This matters most with geometric and mixed series, where a single pair of numbers can accidentally suggest a ratio that the rest of the series does not follow. Two clean confirmations cost barely a second and save you from a wrong answer with negative marking attached, which in a sectional-timed paper is a trade you should always take.