Series: Reasoning Ability Notes for Uttarakhand Police Sub-Inspector (SI)

📚 Reasoning Ability – Series

For Uttarakhand Police Sub‑Inspector (SI) Examination

1️⃣ Simple Explanation

A series is a sequence of numbers (or letters) that follows a particular rule. In reasoning tests, you’re asked to:

Identify the rule (e.g., add 3, multiply by 2, alternate, etc.).
Find the missing term(s) or the next term in the sequence.

Common series types you’ll encounter:

Series Type	Typical Pattern	Example
Arithmetic (A.P.)	Constant difference between consecutive terms.	5, 9, 13, …
Geometric (G.P.)	Constant ratio (multiplication/division).	2, 6, 18, …
Harmonic (H.P.)	Reciprocal of an arithmetic series.	1, ½, ⅓, …
Fibonacci / Recursive	Each term is the sum of previous two.	1, 1, 2, 3, 5, …
Alternating / Mixed	Two or more patterns inter‑leaved.	2, 4, 3, 6, 4, 8, …

Tip: In exams, a single series is usually not a trick; the rule is direct and consistent.
If you can’t spot a rule in 10–12 seconds, the series is probably numeric (not symbolic) and simple (A.P., G.P., or a basic recursive pattern).

2️⃣ Key Points to Remember

#	Point	Why it matters
1	Look for common differences or ratios first.	A.P. / G.P. are the most frequent.
2	Check the first two terms for a hint of the rule.	Many series start with a clear progression.
3	Consider alternating patterns (even‑odd positions).	2, 5, 3, 8, 4, 11,…
4	Look for factorial, powers, or squares.	1, 2, 6, 24, … (n!)
5	Check for recursive relations (sum of previous terms).	Fibonacci, Lucas, etc.
6	Verify consistency with all given terms before deciding.	Avoid false positives.
7	Time‑saving trick: If two consecutive terms differ by a large jump, check for a ratio; if they differ by a small, constant number, check for a difference.	Saves time during the exam.
8	Use “reverse engineering”: If the last term is given, work backwards to find earlier terms.	Useful for “missing term” problems.
9	Always check for “odd‑even” multiplication/addition: e.g., +2, ×3 alternating.	Many exam questions use this.
10	Double‑check the final answer: Plug the answer back into the rule.	Prevents mis‑reading the pattern.

3️⃣ Worked Example (Arithmetic & Mixed)

Problem:
Find the missing term in the series:
4, 9, 14, 19, ?, 29, 34

Solution:

Check differences between consecutive terms:
- 9 – 4 = 5
- 14 – 9 = 5
- 19 – 14 = 5
- ? – 19 = ?
- 29 – ? = ?
- 34 – 29 = 5
The differences are all 5 except for the two involving the missing term.
Deduce the rule:
It appears to be an Arithmetic Progression (A.P.) with common difference 5.
Find the missing term:
- Since 19 + 5 = 24, the missing term must be 24.
- Check the next difference: 29 – 24 = 5 → Consistent.

Answer: 24

Another Example (Geometric & Alternating)

Problem:
Identify the next term in the series:
2, 6, 18, 54, 162, ?

Solution:

Check ratios:
- 6 ÷ 2 = 3
- 18 ÷ 6 = 3
- 54 ÷ 18 = 3
- 162 ÷ 54 = 3
The common ratio is 3 → Geometric Progression.
Next term:
162 × 3 = 486.

Answer: 486

Recursive Pattern Example

Problem:
Find the 5th term of the series:
2, 3, 5, 8, ?

Solution:

Recognize the Fibonacci pattern: each term = sum of previous two.
2 + 3 = 5 → 3rd term.
3 + 5 = 8 → 4th term.
5 + 8 = 13 → 5th term.

Answer: 13

4️⃣ Quick Revision Summary

A.P. (Arithmetic Progression) – Constant difference d.
- Formula: aₙ = a₁ + (n-1)d
- Key: Look for equal differences.
G.P. (Geometric Progression) – Constant ratio r.
- Formula: aₙ = a₁ × rⁿ⁻¹
- Key: Look for equal ratios.
H.P. (Harmonic Progression) – Reciprocal of A.P.
- Formula: 1/aₙ = 1/a₁ + (n-1)/d
- Key: Recognize fractions getting smaller.
Fibonacci / Recursive – Each term = sum (or other function) of previous terms.
- Formula depends on