Figure series: Reasoning Ability Notes for Andaman & Nicobar Islands Police Sub-Inspector (SI)

📚 Study Notes – “Figure Series”

Program – Andaman & Nicobar Islands Police Sub‑Inspector (SI)
Subject – Reasoning Ability
Topic – Figure Series

1. Simple Explanation

A figure series is a sequence of numbers (or shapes) where each term is generated by a rule that is usually systematic and can be observed by comparing consecutive terms. Your job is to identify the rule and predict the next one(s).

Think of it like a riddle hidden in numbers:

The rule could involve arithmetic operations, geometry, patterns of digits, or even external concepts (e.g., calendar dates).
Once you spot the rule, you can solve all the questions in the section quickly.

2. Key Points

#	What to look for	How to spot it
1	Arithmetic Progression (AP)	Constant difference between terms.
2	Geometric Progression (GP)	Constant ratio between terms.
3	Alternating Patterns	Two interleaved sequences (e.g., odd and even positions).
4	Addition/Subtraction of Digits	Sum or difference of digits of the previous term.
5	Multiplication/Division by a constant	Factor applied to the whole number.
6	Increasing/Decreasing by a sequence	Differences themselves follow a pattern (e.g., 1, 3, 5…).
7	Shape/Geometric Progression of Figures	Number of sides, angles, or area.
8	Cyclic or Repeating Patterns	Sequence repeats after a few terms.
9	Non‑numeric patterns	Use of letters, dates, or other symbols.
10	Hidden Sub‑patterns	A rule applies to every second/third term, etc.

Tip:
Always write down the first 4–5 terms. Check for common differences or ratios. If nothing obvious pops up, try splitting the series into two sub‑series (odd & even positions) or look at digit properties.

3. Worked Example

Question:
Find the next term(s) of the series:
3, 7, 15, 31, 63, …

Step‑by‑Step Solution

Term	1	2	3	4	5
Value	3	7	15	31	63

Compute differences:
- 7 – 3 = 4
- 15 – 7 = 8
- 31 – 15 = 16
- 63 – 31 = 32
Observe the differences: 4, 8, 16, 32 → each is doubling the previous one (×2).
This is a geometric progression of differences.
Predict the next difference:
32 × 2 = 64
Add to last term:
63 + 64 = 127

Answer: The next term is 127.

Why this works:
The series is essentially 2ⁿ + 1 for n = 1, 2, 3…
( 2^1 + 1 = 3)
( 2^2 + 1 = 5 ≠ 7) – but if you look carefully:
Actually the pattern is 2ⁿ + 1 starting from 2⁰+1 = 2? – no.
The simpler rule: Start with 3, then keep adding powers of 2 (4, 8, 16, 32, …).

4. Quick Revision Summary

Identify the rule → Look for AP, GP, digit operations, or alternating patterns.
Check differences/ratios → Helps spot AP/GP or hidden patterns.
Split the series → Odd vs even positions, or every 3rd term.
Examine digits → Sum, difference, or product of digits.
Look for doubling/halving → Common in many exam patterns.
Remember cyclicity → Some series repeat after a few terms.
Practice with varied examples → Builds pattern‑recognition speed.

Final Quick Test

Try this series on your own:

8, 12, 20, 36, 68, …

Write down the next term and explain the rule in one sentence.

(Answer hidden for self‑check: 132 – rule: add the previous term’s difference, which doubles each time.)

Good luck! 🎯