Use pattern reasoning to predict a later term.

SECTION 1

Historical Context & Motivation

People have spotted patterns in numbers for thousands of years. Ancient builders in Egypt used patterns to stack stones perfectly. Later, math whizzes like Fibonacci found cool rules in nature, like flower petals. These ideas help us predict what happens next in a list.

3000 BC

Egyptian Pyramids

Workers used repeating patterns to build strong shapes.

1202

Fibonacci Sequence

Italian mathematician spots numbers in pinecones and rabbits.

1800s

Arithmetic Sequences

Math teachers teach kids to add the same number each time.

Today

SSAT Tests

You use patterns to solve quick math puzzles!

Patterns solve the puzzle of 'what's next?' You will see lists on the SSAT. Spot the rule to pick the right answer fast. You can do this!

SECTION 2

Core Principles of Pattern Reasoning

A pattern is a rule that repeats. Look for what changes the same way each time. Common types include adding a fixed number or multiplying by one.

Arithmetic Pattern

Add or subtract the same number each step. Like 2, 5, 8, 11 (add 3).

Geometric Pattern

Multiply by the same number. Like 3, 6, 12, 24 (×2).

Square Numbers

Each term is a number squared. Like 1, 4, 9, 16 (1², 2², 3², 4²).

Shape Patterns

Count shapes that grow. Like triangles: 1, 3, 6, 10.

✦ Key Takeaway

Patterns are like levels in your favorite video game. Each level adds the same challenge. Spot the rule, and you level up to the next term!

SECTION 3

Visualizing Patterns

Arrows show adding 3 each time. The next term is 14.

See the steady jumps? This visual arrow path makes patterns easy to spot. Follow the rule to the end.

SECTION 4

Mathematical Framework

Math gives formulas for patterns. An arithmetic sequence adds a fixed number, called d. Start with a₁ (first term). The nth term is a simple rule.

ARITHMETIC SEQUENCE FORMULA

aₙ = a₁ + (n − 1) × d

aₙ = nth term, n = position, d = common difference (like +3)

GEOMETRIC SEQUENCE FORMULA

aₙ = a₁ × r⁽ⁿ⁻¹⁾

r = common ratio (like ×2)

Plug in numbers step by step. You got this on the SSAT!

SECTION 5

Detailed Pattern Breakdown

Triangles build up: stage 1 (1), stage 2 (3), stage 3 (6), stage 4 (10), stage 5 (15).

Count the extras added each time. This shape growth matches adding 1, then 2, then 3. Predict by seeing the build-up.

SECTION 6

Worked Example

Find the next term: 4, 7, 10, 13, ?

Step-by-Step Solution

Step 1: Check differences7 − 4 = 3, 10 − 7 = 3, 13 − 10 = 3. Add 3 each time.

Common difference d = 3

Step 2: Use formula for 5th terma₅ = 4 + (5 − 1) × 3 = 4 + 12 = 16

Step 3: Verify13 + 3 = 16. Matches!

Pattern confirmed

SECTION 7

Strengths & Comparisons

Compare types to spot the right one fast

Pattern Type	Rule	Example
Arithmetic	Add fixed number	2, 4, 6, 8 (+2)
Geometric	Multiply fixed number	2, 6, 18, 54 (×3)
Squares	n²	1, 4, 9, 16

✦ KEY TAKEAWAY

Like choosing a basketball play: arithmetic for steady steps, geometric for big jumps!

SECTION 8

Connection to Advanced Patterns

Middle Level	Advanced (High School)
Spot by differences or ratios	Infinite sums (series)
Next few terms	General formula aₙ

Master these now for harder tests later. Your pattern skills grow strong!

SECTION 9

Practice Problems

PROBLEM 1 — CONCEPTUAL

What rule fits 1, 3, 5, 7? A) ×2 B) +2 C) +1 D) −2 E) Squares

PROBLEM 2 — BASIC CALCULATION

Next in 5, 10, 15, 20? A) 22 B) 25 C) 30 D) 35 E) 40

PROBLEM 3 — INTERMEDIATE

Next in 2, 4, 8, 16? A) 18 B) 24 C) 32 D) 20 E) 64

PROBLEM 4 — APPLIED

Students: 10, 13, 17, 22. Next? A) 26 B) 27 C) 28 D) 29 E) 30

PROBLEM 5 — CRITICAL THINKING

Shape rows: 1 dot, 4 dots, 9 dots. Next? A) 12 B) 14 C) 16 D) 18 E) 25

SUMMARY

Lesson Summary

Master pattern reasoning by checking differences or ratios. Use formulas like aₙ = a₁ + (n−1)d for steady adds.

Visuals and practice build your skill. You are ready for SSAT success—keep spotting those rules!