Number series for LIC AAO

A number series question gives you five to seven numbers and asks for the next one, or for the one that does not fit. There is no formula to recall and no data to read. It is pure pattern recognition, and most patterns come from a short list. A candidate who checks that list in a fixed order solves most series in 20 to 40 seconds. A candidate who stares at the numbers hoping for inspiration can lose two minutes and still guess.

The checking routine

Work through these checks in order. Stop at the first one that fits every term.

1. How fast does the series grow? Slow growth points to addition or subtraction. Fast growth points to multiplication. A series that rises and falls usually hides two series mixed together.
2. First differences. Write the gap between each pair of terms. Are the gaps constant, rising steadily, squares, cubes, or squares plus or minus one?
3. Second differences. If the first differences are not obvious, take the differences of the differences. A constant second difference means the first differences rise by a fixed amount.
4. Ratios. For fast growth, divide each term by the one before. Look for a constant multiplier, rising multipliers (×2, ×3, ×4), or half-steps (×0.5, ×1.5, ×2.5).
5. Multiply and add. Patterns like ×1 + 1, ×2 + 2, ×3 + 3, or ×2 + 1 each time.
6. Alternate terms. Split the series into odd and even positions and check each on its own.
7. Special numbers. Primes, squares, cubes, and squares or cubes plus or minus one.

Why differences work

If each term is made by adding something to the one before, the differences show you exactly what was added. When the additions themselves follow a pattern, such as 2, 5, 10, 17, the second level of differences reveals it. Most "addition" series in exams become clear within two levels.

Why ratios work

If each term is made by multiplying, the differences grow as fast as the terms and tell you little. Dividing strips the growth out and leaves the multiplier. When a constant is also added, the ratio is close to a whole number but not exactly one. That tells you to look for "× n + something".

Useful numbers to know

The primes below 50 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43 and 47.

Worked examples

Example 1. 7, 9, 14, 24, 41, ?
Growth is slow, so take differences: 2, 5, 10, 17.
These are 1 + 1, 4 + 1, 9 + 1, 16 + 1, squares plus one.
The next difference is 25 + 1 = 26, so the answer is 41 + 26 = 67.

Example 2. 5, 6, 14, 45, 184, ?
Growth speeds up quickly, so try ratios. 6/5 is a little over 1, 14/6 a little over 2, 45/14 a little over 3.
Test ×n + n: 5 × 1 + 1 = 6, 6 × 2 + 2 = 14, 14 × 3 + 3 = 45, 45 × 4 + 4 = 184.
Next: 184 × 5 + 5 = 920 + 5 = 925.

Example 3. 12, 18, 30, 50, 80, ?
First differences: 6, 12, 20, 30. Second differences: 6, 8, 10.
The second differences rise by 2, so the next is 12 and the next first difference is 30 + 12 = 42.
Answer: 80 + 42 = 122. The first differences are also 2 × 3, 3 × 4, 4 × 5, 5 × 6, and 6 × 7 = 42 confirms it.

Example 4. 64, 32, 48, 120, 420, ?
The series falls, then rises fast, so check ratios: 32/64 = 0.5, 48/32 = 1.5, 120/48 = 2.5, 420/120 = 3.5.
The multiplier rises by 1 each time. Next: 420 × 4.5 = 1,890.

Example 5. 3, 20, 6, 17, 12, 14, 24, ?
The terms rise and fall, so split them.
Odd positions: 3, 6, 12, 24, doubling.
Even positions: 20, 17, 14, falling by 3.
The missing term is in an even position (the eighth), so it is 14 − 3 = 11.

Example 6 (wrong term). 4, 6, 12, 30, 90, 316, 1,260
Ratios: 1.5, 2, 2.5, 3, then 316/90 ≈ 3.51, then 1,260/316 ≈ 3.99.
The pattern is ×1.5, ×2, ×2.5, ×3, ×3.5, ×4. So the fifth step should give 90 × 3.5 = 315, and 315 × 4 = 1,260 fits the last term.
The wrong term is 316.

Missing-term and wrong-term questions compared

A time rule

Give each series about 30 to 40 seconds. If none of the seven checks fits by then, move on and come back at the end. A second look often shows the pattern at once, because you are no longer fixed on the first idea you tried.

Practice set

1. 11, 12, 16, 25, 41, ? 66. Differences 1, 4, 9, 16 are squares, so the next is 25.
2. 2, 3, 8, 27, 112, ? 565. ×1 + 1, ×2 + 2, ×3 + 3, ×4 + 4, then 112 × 5 + 5 = 565.
3. 2, 10, 37, 101, 226, ? 442. Differences 8, 27, 64, 125 are cubes, so the next is 216, and 226 + 216 = 442.
4. 5,040, 2,520, 840, 210, 42, ? 7. ÷2, ÷3, ÷4, ÷5, then 42 ÷ 6 = 7.
5. 17, 19, 23, 29, 31, ? 37. Consecutive prime numbers.
6. 9, 15, 27, 51, 99, ? 195. Each term is ×2 − 3: 99 × 2 − 3 = 195. The differences 6, 12, 24, 48 double, which confirms it.
7. Find the wrong term: 6, 7, 15, 46, 184, 926. 184. The pattern is ×1 + 1, ×2 + 1, ×3 + 1, ×4 + 1, ×5 + 1. 46 × 4 + 1 = 185, and 185 × 5 + 1 = 926.
8. Find the wrong term: 8, 12, 20, 36, 70, 132. 70. The differences should double: 4, 8, 16, 32, 64. So the fifth term is 36 + 32 = 68, and 68 + 64 = 132.

A one-week series drill

What to do next

• Learn squares to 30 and cubes to 15 so they jump out of a series.
• Solve ten series a day for two weeks, timing each block.
• Add quadratic comparisons and simplification, the other quick-scoring prelims blocks.
• See the whole section plan in a quant plan for LIC AAO.