Series completion is one of those topics where speed is almost entirely a function of pattern vocabulary — how many rule-types you have stored and how fast you can match the series in front of you to one of them.
The idea is simple: you are given a sequence of numbers, letters, or alphanumeric terms with one term missing (usually at the end, occasionally in the middle), and you have to identify the underlying rule to fill in the gap.
Here's an analogy that works well. Think of a series like a lock combination. Once you know the mechanism — turn right three clicks, left two, right one — you can open it every time. The series question is asking: what is the mechanism? Once you name it, the answer falls out in seconds.
What makes AFCAT series questions tractable is that the underlying mechanisms are not exotic. The paper consistently tests a small repertoire of patterns: arithmetic progressions, geometric progressions, doubling or halving with a constant added, perfect squares or cubes, and arithmetic progressions of differences (second-order sequences). You will rarely encounter anything more complex in AFCAT.
The practical challenge is not the math — it is recognizing the pattern type before you waste time trying the wrong approach. A student who tries to compute differences first on every series will lose time on a series that is purely geometric. The goal of this page is to build a mental triage system so you look at a series and immediately know which family it belongs to.
One more thing before we go deeper: in AFCAT, series questions tend to be clean. The numbers involved are not deliberately ugly. If your working produces a messy decimal or a non-integer, you have almost certainly picked the wrong pattern. Trust that signal and switch approaches.
Every series you will encounter in AFCAT belongs to one (or at most two combined) of these families. Learn to name them out loud as you read the series — that naming habit is what separates a 12-second solve from a 45-second grind.
1. Arithmetic Progression (AP)
Each term is obtained by adding (or subtracting) a fixed constant d to the previous term.
a, a+d, a+2d, a+3d, ...
Recognition signal: compute first differences (term2 − term1, term3 − term2, etc.). If they are all equal, it is a straight AP. Example: 5, 11, 17, 23 — differences are all 6.
2. Geometric Progression (GP)
Each term is obtained by multiplying the previous term by a fixed ratio r.
a, ar, ar², ar³, ...
Recognition signal: compute ratios (term2 / term1, term3 / term2). If they are equal, it is GP. Example: 3, 6, 12, 24 — ratio is 2 throughout. The series 2, 6, 18, 54 has ratio 3.
3. Doubling / Halving with an Additive Constant
The rule is next term = (current term × k) + c where k is typically 2 and c is a small integer (often 1 or 2).
Recognition signal: try multiplying by 2 and see if you are always off by the same amount. Example: 3, 8, 18, 38 — multiply by 2: 6, 16, 36... you are always 2 short of the given term, so the rule is ×2 + 2.
This family comes up repeatedly in AFCAT. The two most common variants are ×2 + 1 and ×2 + 2. Keep both on your radar.
4. Perfect Squares / Cubes
The terms are 1², 2², 3², ... or 1³, 2³, 3³, .... These are among the easiest to recognise visually because the numbers are familiar: 1, 4, 9, 16, 25, 36, 49 for squares and 1, 8, 27, 64, 125 for cubes.
5. Arithmetic Differences (Second-Order AP)
The differences between consecutive terms themselves form an AP. Example: 100, 96, 88, 76 — differences are -4, -8, -12 (themselves an AP with common difference -4). The next difference is -16, giving 76 − 16 = 60.
Recognition signal: first differences are not constant, but second differences (differences of differences) are constant.
6. Sum of Previous Terms
term(n) = term(n-1) + term(n-2). The Fibonacci-type pattern. Recognise it when you see that each term is roughly the sum of the two before it.
7. Mixed / Alternating Patterns
Two interleaved series running in alternate positions. Extract odd-indexed and even-indexed terms separately; each sub-series will follow one of the simpler rules above.
When you look at a series, always compute first differences as your first move — even before you attempt to recognise the pattern. This takes three subtraction steps and costs you about four seconds. The result tells you immediately which family you are in:
Do not skip this step. Students who try to guess the pattern visually waste time when the pattern is second-order.
AFCAT also tests letter series. The core principle is identical — assign positional values (A=1, B=2, ... Z=26) and then apply the number-series logic to the positions.
Look for: constant step patterns (A, D, G, J — every third letter), alternating step patterns, or alphabetical groups. When a letter series has three-letter groups (e.g., ABD, CFI, DJP), analyse position-by-position within each group.
Always verify your answer by checking that the pattern holds across all given terms, not just the last two.
Before attempting any series, write out first differences in a row below the series. For a 5-term series, this takes roughly 4 seconds and four subtractions. If those differences are constant, you are done (AP). If not, write second differences below the first — another 3 seconds. If second differences are constant, it is a second-order sequence and your next term's difference is predictable. This two-row check eliminates guesswork on roughly 60% of series you will see. Standard approach (trial-and-error across multiple rules): 30-40 seconds. Difference-table approach: 8-12 seconds for the pattern identification step alone.
When first differences are increasing but not in a constant additive pattern, immediately try multiplying each term by 2. If the result is consistently c more or less than the next term, the rule is ×2 + c. For AFCAT, the two most frequent values of c are +1 and +2. Test both in sequence. Micro-example: series 1, 3, 7, 15 — multiply each by 2: 2, 6, 14 — each is 1 less than the next term, confirming ×2 + 1. This check takes 6 seconds vs. 25-30 seconds trying multiple patterns sequentially.
When a series contains familiar-looking numbers (1, 4, 9, 16, 25 or numbers close to them), immediately take the square root of each term mentally. If the roots are consecutive integers, it is a perfect-square series and the next term is (n+1)². No difference computation needed. Example: 1, 4, 9, 16, 25 — roots are 1, 2, 3, 4, 5 — next root is 6, next term is 36. Standard method (computing differences, then recognising the growing-difference pattern): 20 seconds. Square-root recognition: 5 seconds.
When you identify a second-order sequence (differences are themselves an AP), compute what the next difference must be before looking at the options. In AFCAT, the options are usually spaced far enough apart that one quick computation eliminates three options immediately. Example: 100, 96, 88, 76 — differences are -4, -8, -12, so next difference is -16, giving 60. Glance at options: 60, 64, 68, 72 — only one matches. You spend zero time verifying the wrong options. This cuts average time from 25 seconds to 14 seconds for second-order questions.
For any series where terms are growing fast, divide the second term by the first. If that ratio equals third-divided-by-second and fourth-divided-by-third, it is a clean GP. Do all three divisions only if the first one gives a whole number. If the first division gives a fraction, abandon this check and move to the ×2+c approach instead. Micro-example: 7, 14, 28, 56 — 14/7=2, 28/14=2, confirmed GP with r=2. Next term: 56×2=112. Three divisions in about 5 seconds vs. writing out differences: saves roughly 10 seconds and avoids the misleading first-difference sequence (7, 14, 28) which looks like a GP in differences but could confuse you if you approach it as a second-order sequence.
Use this decision tree in the exam hall. Do not deviate from the order — it is optimised for the pattern frequency in AFCAT.
Step 1 — Are the terms familiar squares or cubes? If yes (1, 4, 9, 16, 25 or 1, 8, 27, 64), name the pattern and write the next term immediately.
Step 2 — Compute first differences.
Step 3 — If first differences do not fit Steps 1 or 2, try the ×2+c check. Multiply last four terms by 2 and see if the gap to the next term is constant. Test c = +1, then c = +2, then c = −1.
Step 4 — If still unclear, check the ratio (GP test). Divide consecutive terms. If ratio is constant, extend by multiplication.
Step 5 — If the series has 6+ terms, check for alternating sub-series. Extract odd and even positions separately and apply Steps 1-4 to each.
If you have not identified the pattern within 25 seconds, use option elimination: compute what is plausible given the growth trend and eliminate outliers.
Why this question: The ×2 + 2 family is one of the two most frequently tested compound-operation patterns in AFCAT. Being able to spot it in under 10 seconds is a mark-differentiator.
Solving path: Compute first differences: 5, 10, 20. These are doubling, not constant — so it is not a plain AP. Move to ×2 check: 3×2=6, but next term is 8 (off by +2). Test consistently: 8×2=16, next is 18 (+2). 18×2=36, next is 38 (+2). Rule confirmed: multiply by 2, add 2. Apply: 38×2+2 = 78.
Why this question: The ×2 + 1 family looks very similar to ×2 + 2. AFCAT uses both to test whether you are checking carefully or guessing from the first match. One extra second of verification saves you from a wrong answer.
Solving path: Differences: 2, 4, 8, 16 — each doubling. This is a sign of a compound operation, not plain differences. Check ×2: 1×2=2, next is 3 (off by +1). Consistent across all terms. Rule: ×2+1. Apply: 31×2+1 = 63.
Why this question: The perfect-square series is the fastest-to-solve type in AFCAT, but only if you recognise it immediately. If you fall into computing differences instead, you lose time on the easiest question on the paper.
Solving path: Look at the terms: 1, 4, 9, 16, 25. Recognition is instant if you have drilled squares up to 15. Roots: 1, 2, 3, 4, 5. Next root: 6. Next term: 36. Total time: under 6 seconds.
Why this question: The second-order AP (arithmetic progression of differences) is the most commonly fumbled series type. Students compute first differences, see they are not constant, and panic. Calm down — compute second differences next.
Solving path: First differences: 96−100=−4, 88−96=−8, 76−88=−12. Not constant. Second differences: −8−(−4)=−4, −12−(−8)=−4. Constant at −4. Next first difference: −12+(−4)=−16. Next term: 76+(−16) = 60.
Why this question: This is a clean GP with ratio 3. The trap is that students sometimes try to compute differences (4, 12, 36) and then notice those are also a GP with ratio 3 — which can lead to confusion about whether to extend the original series or the difference series. Always extend the original series using the ratio, not the difference series.
Solving path: Compute ratio: 6/2=3, 18/6=3, 54/18=3. Confirmed GP with r=3. Next term: 54×3=162.
Stopping at first differences without checking second differences. When first differences are not constant, many candidates give up on the AP hypothesis and randomly try other approaches. Always check second differences before moving on — it takes three extra subtractions.
Confusing ×2+1 with ×2+2. The two patterns feel similar when you are reading quickly. Always verify the additive constant across all consecutive pairs, not just the first one. A single-term check will get you the wrong answer when the pattern is ×2+2 and you assumed ×2+1.
Treating a GP as an AP with large differences. When terms double or triple rapidly (7, 14, 28, 56), first differences (7, 14, 28) also grow rapidly. Some candidates compute these differences and try to extend a "difference series" instead of simply multiplying by the ratio. If the ratio of consecutive terms is constant, use the ratio directly.
Not verifying the rule across all given terms. AFCAT options are designed so that one wrong assumption about the pattern gives a plausible-but-wrong answer. You cannot verify using only the last two terms. Check that your identified rule holds from term 1 to term 2, term 2 to term 3, and so on.
Ignoring the square/cube recognition shortcut. Series like 1, 4, 9, 16, 25 should take under 6 seconds. Candidates who do not have their squares memorised spend 20 seconds computing differences (1, 2, 3, 4 — wait, these are the roots themselves, so the second differences would be all 1s). Memorise squares up to at least 15² = 225 and cubes up to 10³ = 1000.
Misreading a negative difference as no difference. In decreasing series (100, 96, 88, 76), the differences are negative. Candidates sometimes write absolute values of differences and then lose track of the sign when applying the rule to find the next term. Always carry the sign explicitly.