Series

Series problems show 4–6 terms and ask you to predict the next. The pattern is usually arithmetic, geometric, or a combination.

Arithmetic difference (constant gap): 4, 7, 10, 13, … (add 3).
Geometric ratio (constant multiplier): 3, 6, 12, 24, … (×2).
Squares / cubes: 1, 4, 9, 16, 25 (n²); 1, 8, 27, 64 (n³).
Triangular / pentagonal: 1, 3, 6, 10, 15 (n(n+1)/2).
Differences-of-differences (second-order): 2, 5, 10, 17, 26 — differences 3,5,7,9 are themselves arithmetic.
Alternating series: 2, 5, 4, 9, 6, 13 — odd-position terms (2, 4, 6) and even-position (5, 9, 13) form two separate series.
Mixed-operation: ×n, +n alternating (5, 10, 13, 26, 29, 58, …: ×2, +3, ×2, +3, ×2).
Fibonacci-style: each term = sum of previous two: 1, 1, 2, 3, 5, 8, 13.

Letter series: substitute letter ↔ position (A=1, B=2, … Z=26). Then it's just a number series.
Example: B, D, F, H, ?, L → +2 each → next is J.

Alphanumeric (letter + number combo): A1, B4, C9, D16, ? → letters A,B,C,D (+1); numbers 1,4,9,16 (squares). Next: E25 → E25.

Spotting patterns — checklist:

1. Subtract consecutive terms — is the difference constant or in pattern?
2. Divide consecutive terms — is the ratio constant?
3. Check squares/cubes near each term.
4. Look at digit sums or last digits.
5. Split into two interleaved series if positions feel "alternating".

Example 1: 7, 11, 19, 35, ?
Differences: 4, 8, 16 (doubling). Next diff = 32. Answer: 35 + 32 = 67.

Example 2: 1, 4, 27, 256, ?
Pattern: 1¹, 2², 3³, 4⁴, so next = 5⁵ = 3125.

Example 3: 2, 3, 5, 7, 11, 13, ?
Sequence of primes. Next prime after 13 = 17.

Example 4: 5, 7, 12, 19, 31, 50, ?
Each term = sum of previous two. 19 + 31 = 50 ✓. Next: 31 + 50 = 81.

Example 5: 6, 13, 27, 55, 111, ?
Pattern: each ≈ ×2 + something. 6×2+1=13. 13×2+1=27. 27×2+1=55. 55×2+1=111. So next: 111×2+1 = 223.

Example 6 (letter series): A, C, F, J, ?
Differences in alphabet: +2, +3, +4 → next +5. J+5 = O.

Example 7 (alphanumeric): A4, D9, G16, J25, ?
Letters: A, D, G, J (gap 3) → next M. Numbers: 4, 9, 16, 25 (squares of 2,3,4,5) → next 36. Answer: M36.

Speed tactics:

• For 5–6 term series, finding the pattern in the first 3 differences usually suffices.
• If differences fail, try ratios.
• For "wrong term" questions (which term doesn't fit?), continue the pattern and compare against the given term.
• When confused, check option list — extreme values often suggest exponential growth; close values suggest arithmetic.

Common traps: confusing Fibonacci-style with "differences double" (5, 8, 13, 21, … sums vs 7, 11, 19, 35, … doubled diffs).