9 min readConcept sheet2-mark items; 5- and 10-mark 'types of series with examples'

Number and letter series

A series follows one hidden rule. Find the differences first; if they do not settle, try ratios, squares, cubes, alternate terms and letter positions.

First testDifferences between terms, then differences of differences

Second testRatios: ×2, ×3, or mixed rules such as ×2 + 1

Letter positionsA = 1 … Z = 26; E, J, O, T, Y = 5, 10, 15, 20, 25

Wrong-term typeFind the one term that breaks the rule

Common patterns

Pattern	Example	Rule
Constant difference	7, 12, 17, 22, 27	+5
Growing difference	2, 5, 10, 17, 26	+3, +5, +7, +9 (n² + 1)
Constant ratio	3, 6, 12, 24, 48	×2
Mixed rule	2, 5, 11, 23, 47	×2 + 1
Squares / cubes	1, 4, 9, 16, 25 / 1, 8, 27, 64	n² / n³
Alternate series	2, 10, 4, 20, 6, 30	2, 4, 6 and 10, 20, 30 in turn
Primes	2, 3, 5, 7, 11, 13	prime numbers in order
Letter skip	B, E, H, K, N	+3 places
Letter pairs	AZ, BY, CX, DW	first letter +1, second −1

How to crack a series

1
Look at the size: slow growth suggests + or −; fast growth suggests × or powers.
2
Write the differences under the terms. If they are equal, the rule is found.
3
If not, take the differences of the differences, or look for a pattern in them (odd numbers, squares, doubling).
4
Try ratios, including mixed rules such as ×2 + 1 or ×3 − 1.
5
Check alternate terms for two series woven together.
6
For letters, write positions (A = 1 … Z = 26) above each letter and treat them as numbers.
7
Confirm the rule on every given term before you fill the blank.

Worked example 1: next term from differences

Problem

Find the next term: 6, 11, 21, 36, 56, ?

Working

Standard method: differences are 5, 10, 15, 20. The next difference is 25, so the next term is 56 + 25 = 81.
Shortcut: the differences are 5 × 1, 5 × 2, 5 × 3 …, so the 6th term = 6 + 5 × (1 + 2 + 3 + 4 + 5) = 6 + 5 × 15 = 81.
Check: the same formula gives the 4th term as 6 + 5 × 6 = 36, which matches.

Answer

81.

Worked example 2: find the wrong term

Problem

Find the wrong term: 3, 7, 15, 31, 64, 127.

Working

Standard method: differences are 4, 8, 16, 33, 63. They should double: 4, 8, 16, 32, 64. The break is at 64, which should be 63 (then the differences are 32 and 64).
Shortcut: each term = previous × 2 + 1. 31 × 2 + 1 = 63, not 64; and 63 × 2 + 1 = 127 confirms the rule.
Check: every correct term is one less than a power of 2: 3, 7, 15, 31, 63, 127.

Answer

64 is wrong; it should be 63.

Worked example 3: letter series

Problem

Find the next term: AC, FH, KM, PR, ?

Working

Step 1: write positions: AC = 1, 3; FH = 6, 8; KM = 11, 13; PR = 16, 18.
Step 2: first letters go 1, 6, 11, 16 (+5), so next is 21 = U. Second letters go 3, 8, 13, 18 (+5), so next is 23 = W.
Check: inside every pair the gap is 2 places (A_C, F_H, K_M, P_R, U_W).

Answer

UW.

How exams ask this

Bank papers set missing-number and wrong-number series of about five questions; SSC and railway papers mix number, letter and alphanumeric series. Placement tests favour mixed rules (×2 + 1) and alternate series, so check differences, ratios and alternate terms in that order.

Exam angle

2 marks: “Find the next term: 2, 6, 12, 20, 30, ?” 42. The terms are n(n + 1): 1 × 2, 2 × 3, … 6 × 7; the differences are 4, 6, 8, 10, 12.

5 marks: “Explain any five types of number series with an example each.” Constant difference, growing difference, ratio, mixed rule, squares or cubes and alternate series, each with a short example and its rule.

10 marks: “Explain number and letter series with their types and the method of solving, with examples.” Define; draw the D-R-P-A method; number types; letter series with positions; wrong-term series; one fully worked example.

Remember it

D-R-P-A: Differences, Ratios, Powers (squares, cubes), Alternate terms. Letters: EJOTY = 5, 10, 15, 20, 25.