3.2 Sequences, Patterns & Series Calculations

Armed Forces Qualification Test AFQT

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1 Chapter 1: AFQT Structure, Scoring & Enlistment Requirements

1.1 AFQT Overview, Subtest Breakdown & CAT-ASVAB vs. P&P Format 1.2 The AFQT Scoring Formula (2×VE + AR + MK) & Percentile Norming 1.3 Branch Minimum Scores, AFQT Categories & Retake Policies

2 Chapter 2: Arithmetic Reasoning (AR) — Ratios, Rates & Percentages

2.1 Ratios, Proportions & Scale Conversions 2.2 Percentages, Discounts, Interest & Tax Calculations 2.3 Rate, Distance, Time & Speed Word Problems 2.4 Work Rates & Combined Labor Problems

3 Chapter 3: Arithmetic Reasoning (AR) — Mixtures, Sequences & Applied Math

3.1 Mixture, Concentration & Price-Blending Problems 3.2 Sequences, Patterns & Series Calculations 3.3 Averages, Weighted Averages & Statistical Word Problems

4 Chapter 4: Word Knowledge (WK) — Vocabulary & Word Structure

4.1 High-Frequency AFQT Vocabulary & Synonym Recognition 4.2 Latin & Greek Roots, Prefixes & Suffixes for Word Decoding 4.3 Context Clues & Words with Multiple Meanings

5 Chapter 5: Paragraph Comprehension (PC) — Reading Analysis & Inference

5.1 Identifying Main Ideas & Primary Purpose 5.2 Locating Supporting Details & Direct Facts 5.3 Drawing Logical Inferences & Implied Conclusions 5.4 Analyzing Tone, Author Perspective & Contextual Vocabulary

6 Chapter 6: Mathematics Knowledge (MK) — Fundamentals & Linear Algebra

6.1 Arithmetic Operations, Order of Operations & Number Properties 6.2 Exponents, Scientific Notation & Square Roots / Radicals 6.3 Linear Equations, Inequalities & Absolute Value 6.4 Systems of Linear Equations & Word-to-Algebra Translation

7 Chapter 7: Mathematics Knowledge (MK) — Advanced Algebra & Functions

7.1 Polynomial Operations & Factoring Techniques 7.2 Quadratic Equations, Factoring & Quadratic Formula 7.3 Coordinate Geometry: Slope, Distance, Midpoint & Linear Graphs

8 Chapter 8: Mathematics Knowledge (MK) — Geometry & Spatial Math

8.1 Angles, Triangles & Pythagorean Theorem 8.2 Quadrilaterals, Polygons & Circle Geometry 8.3 3D Geometry: Surface Area & Volume of Solids 8.4 Basic Geometric Transformations & Trigonometric Ratios

3.2 Sequences, Patterns & Series Calculations

Key Takeaways Explain This Topic AI

Arithmetic sequences progress by adding or subtracting a constant common difference (d), defined by the nth term formula a_n = a_1 + (n - 1)d.

Geometric sequences progress by multiplying or dividing by a constant common ratio (r), defined by the nth term formula a_n = a_1 * r^(n-1).

The sum of an arithmetic series is calculated instantaneously using Gauss's formula S_n = (n / 2) * (a_1 + a_n), bypassing tedious manual addition.

Second-order (quadratic) sequences feature first-level differences that change at a constant rate, requiring second-level difference analysis to determine the next term.

Alternating and interleaved series contain two distinct sub-sequences combined into odd and even position indices, requiring split-pattern tracking.

Last updated: July 2026

3.2 Sequences, Patterns & Series Calculations

Quick Answer: A numerical sequence is an ordered list of numbers following a specific mathematical rule. In arithmetic sequences , terms change by a constant added difference d d d ( a n = a 1 + ( n − 1 ) d a_n = a_1 + (n-1)d a n ​ = a 1 ​ + ( n − 1 ) d ). In geometric sequences , terms change by a constant multiplier ratio r r r ( a n = a 1 ⋅ r n − 1 a_n = a_1 \cdot r^{n-1} a n ​ = a 1 ​ ⋅ r n − 1 ). To quickly sum an arithmetic series, use S n = n 2 ( a 1 + a n ) S_n = \frac{n}{2}(a_1 + a_n) S n ​ = 2 n ​ ( a 1 ​ + a n ​ ) .

Sequence and series problems on the AFQT evaluate your ability to identify inductive patterns, abstract mathematical rules, and apply linear and exponential models. Questions range from finding the immediate next term in a complex sequence to calculating the total cumulative sum of an extended series over dozens of iterations.

Arithmetic Sequences (Linear Growth & Decay)

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This constant value is called the common difference ( d d d ) .

Key Arithmetic Formulas Common Difference: d = a k + 1 − a k \text{Common Difference: } d = a_{k+1} - a_k Common Difference: d = a k + 1 ​ − a k ​ General Term Formula: a n = a 1 + ( n − 1 ) d \text{General Term Formula: } a_n = a_1 + (n - 1)d General Term Formula: a n ​ = a 1 ​ + ( n − 1 ) d

Where:

• a n a_n a n ​ = the value of the n n n -th term
• a 1 a_1 a 1 ​ = the first term in the sequence
• n n n = the position number of the target term
• d d d = the common difference
• Step-by-Step Worked Example: Finding an Advanced Term

Problem: A logistics unit receives supply shipments on a schedule where the crate count increases by a constant amount each shipment. Shipment 1 contains 45 crates, Shipment 2 contains 58 crates, and Shipment 3 contains 71 crates. How many crates will be delivered in Shipment 18?

• Step 1: Identify the Sequence Type and Variables
• a 1 = 45 a_1 = 45 a 1 ​ = 45
• a 2 = 58 a_2 = 58 a 2 ​ = 58
• a 3 = 71 a_3 = 71 a 3 ​ = 71

Common difference d = 58 − 45 = 13 d = 58 - 45 = 13 d = 58 − 45 = 13 (and 71 − 58 = 13 71 - 58 = 13 71 − 58 = 13 )

Target term position n = 18 n = 18 n = 18

Step 2: Apply the General Term Formula a 18 = a 1 + ( 18 − 1 ) d a_{18} = a_1 + (18 - 1)d a 18 ​ = a 1 ​ + ( 18 − 1 ) d a 18 = 45 + ( 17 × 13 ) a_{18} = 45 + (17 \times 13) a 18 ​ = 45 + ( 17 × 13 )

Step 3: Compute the Multiplication and Final Sum 17 × 13 = 221 17 \times 13 = 221 17 × 13 = 221 a 18 = 45 + 221 = 266 crates a_{18} = 45 + 221 = 266 \text{ crates} a 18 ​ = 45 + 221 = 266 crates

Shipment 18 will deliver 266 crates .

Arithmetic Series Summation ( S n S_n S n ​ )

When an AFQT question asks for the total accumulation or total sum of an arithmetic sequence over n n n steps, adding every individual term manually wastes valuable test time and introduces calculation errors. Use Gauss's Summation Formula : S n = n 2 ( a 1 + a n ) or S n = n 2 [ 2 a 1 + ( n − 1 ) d ] S_n = \frac{n}{2} (a_1 + a_n) \quad \text{or} \quad S_n = \frac{n}{2} [2a_1 + (n - 1)d] S n ​ = 2 n ​ ( a 1 ​ + a n ​ ) or S n ​ = 2 n ​ [ 2 a 1 ​ + ( n − 1 ) d ]

Step-by-Step Worked Example: Total Cumulative Accumulation

Problem: A recruit begins a physical conditioning program by performing 12 push-ups on Day 1, 15 push-ups on Day 2, and 18 push-ups on Day 3, adding 3 push-ups every single day. What is the total cumulative number of push-ups performed by the recruit over 30 days?

• Step 1: Extract Given Parameters
• a 1 = 12 a_1 = 12 a 1 ​ = 12
• d = 3 d = 3 d = 3
• n = 30 n = 30 n = 30

Step 2: Determine the Final Day's Value ( a 30 a_{30} a 30 ​ ) a 30 = a 1 + ( 30 − 1 ) d = 12 + ( 29 × 3 ) = 12 + 87 = 99 push-ups a_{30} = a_1 + (30 - 1)d = 12 + (29 \times 3) = 12 + 87 = 99 \text{ push-ups} a 30 ​ = a 1 ​ + ( 30 − 1 ) d = 12 + ( 29 × 3 ) = 12 + 87 = 99 push-ups

Step 3: Calculate the Series Sum ( S 30 S_{30} S 30 ​ ) S 30 = 30 2 ( a 1 + a 30 ) S_{30} = \frac{30}{2} (a_1 + a_{30}) S 30 ​ = 2 30 ​ ( a 1 ​ + a 30 ​ ) S 30 = 15 × ( 12 + 99 ) S_{30} = 15 \times (12 + 99) S 30 ​ = 15 × ( 12 + 99 ) S 30 = 15 × 111 = 1 , 665 push-ups S_{30} = 15 \times 111 = 1,665 \text{ push-ups} S 30 ​ = 15 × 111 = 1 , 665 push-ups

The recruit performs a total of 1,665 push-ups over 30 days.

Geometric Sequences (Exponential Growth & Decay)

A geometric sequence is a sequence where each term after the first is found by multiplying the previous term by a non-zero constant called the common ratio ( r r r ) .

Key Geometric Formulas Common Ratio: r = a k + 1 a k \text{Common Ratio: } r = \frac{a_{k+1}}{a_k} Common Ratio: r = a k ​ a k + 1 ​ ​ General Term Formula: a n = a 1 ⋅ r n − 1 \text{General Term Formula: } a_n = a_1 \cdot r^{n-1} General Term Formula: a n ​ = a 1 ​ ⋅ r n − 1

Step-by-Step Worked Example: Exponential Growth

Problem: A cyber defense system detects a malware infection that spreads exponentially across server clusters. At Hour 1, 4 servers are infected. At Hour 2, 12 servers are infected. At Hour 3, 36 servers are infected. If this pattern continues uninterrupted, how many servers will be infected at Hour 6?

• Step 1: Identify Sequence Parameters
• a 1 = 4 a_1 = 4 a 1 ​ = 4
• a 2 = 12 a_2 = 12 a 2 ​ = 12

Common ratio r = 12 4 = 3 r = \frac{12}{4} = 3 r = 4 12 ​ = 3 (verifying: 36 12 = 3 \frac{36}{12} = 3 12 36 ​ = 3 )

Target hour n = 6 n = 6 n = 6

Step 2: Substitute into Geometric Formula a 6 = a 1 ⋅ r 6 − 1 a_6 = a_1 \cdot r^{6-1} a 6 ​ = a 1 ​ ⋅ r 6 − 1 a 6 = 4 ⋅ 3 5 a_6 = 4 \cdot 3^5 a 6 ​ = 4 ⋅ 3 5

Step 3: Evaluate Exponent and Multiply 3 5 = 3 × 3 × 3 × 3 × 3 = 243 3^5 = 3 \times 3 \times 3 \times 3 \times 3 = 243 3 5 = 3 × 3 × 3 × 3 × 3 = 243 a 6 = 4 × 243 = 972 servers a_6 = 4 \times 243 = 972 \text{ servers} a 6 ​ = 4 × 243 = 972 servers

At Hour 6, exactly 972 servers will be infected.

Second-Order (Quadratic) Sequences

In a second-order sequence , the first-level differences between consecutive terms are not constant, but the differences of the differences (second differences) are constant.

Diagnostic Method for Unfamiliar Patterns

When faced with an unfamiliar numerical sequence on the AFQT, construct a difference tree: Given Terms: 3 7 13 21 31 43 1st Differences: + 4 + 6 + 8 + 10 + 12 2nd Differences: + 2 + 2 + 2 + 2 \begin{array}{lccccccccccc} \text{Given Terms:} & 3 & & 7 & & 13 & & 21 & & 31 & & 43 \\ \text{1st Differences:} & & +4 & & +6 & & +8 & & +10 & & +12 & \\ \text{2nd Differences:} & & & +2 & & +2 & & +2 & & +2 & & \end{array} Given Terms: 1st Differences: 2nd Differences: ​ 3 ​ + 4 ​ 7 + 2 ​ + 6 ​ 13 + 2 ​ + 8 ​ 21 + 2 ​ + 10 ​ 31 + 2 ​ + 12 ​ 43 ​

Finding the Next Term

The 1st differences ( 4 , 6 , 8 , 10 , 12 4, 6, 8, 10, 12 4 , 6 , 8 , 10 , 12 ) increase by + 2 +2 + 2 each step.

The next 1st difference must be 12 + 2 = 14 12 + 2 = 14 12 + 2 = 14 .

Add this difference to the last term: 43 + 14 = 57 43 + 14 = 57 43 + 14 = 57 .

Interleaved and Alternating Sequences

Some AFQT pattern questions combine two separate sequences into a single alternating list.

Example of Interleaved Pattern 5 , 100 , 10 , 95 , 15 , 90 , 20 , ? ‾ \mathbf{5}, 100, \mathbf{10}, 95, \mathbf{15}, 90, \mathbf{20}, \underline{?} 5 , 100 , 10 , 95 , 15 , 90 , 20 , ? ​

Analysis: Split into Odd and Even Indices

Odd Terms (Positions 1, 3, 5, 7): 5 , 10 , 15 , 20 … 5, 10, 15, 20 \dots 5 , 10 , 15 , 20 … (Increases by + 5 +5 + 5 )

Even Terms (Positions 2, 4, 6, 8): 100 , 95 , 90 , ? ‾ 100, 95, 90, \underline{?} 100 , 95 , 90 , ? ​ (Decreases by − 5 -5 − 5 )

The 8th term is at an even index, so it follows the even pattern: 90 − 5 = 85 90 - 5 = 85 90 − 5 = 85 .

Sequence Formula Reference Summary

Sequence Type Core Property General Term Formula ( a n a_n a n ​ ) Key AFQT Metric

Arithmetic Constant Difference ( d d d ) a n = a 1 + ( n − 1 ) d a_n = a_1 + (n-1)d a n ​ = a 1 ​ + ( n − 1 ) d Sum: S n = n 2 ( a 1 + a n ) S_n = \frac{n}{2}(a_1 + a_n) S n ​ = 2 n ​ ( a 1 ​ + a n ​ )

Geometric Constant Ratio ( r r r ) a n = a 1 ⋅ r n − 1 a_n = a_1 \cdot r^{n-1} a n ​ = a 1 ​ ⋅ r n − 1 Exponential growth/decay factor

Quadratic Constant 2nd Difference a n = A n 2 + B n + C a_n = A n^2 + B n + C a n ​ = A n 2 + B n + C 1st difference grows linearly

• Interleaved Dual Sub-sequences Separate odd/even rules Track index positions independently
• AFQT Exam Strategy & Rapid Pattern Recognition

First Check Differences: Always test for a constant first difference before assuming a complex rule.

Second Check Ratios: If differences expand rapidly ( 2 , 6 , 18 , 54 2, 6, 18, 54 2 , 6 , 18 , 54 ), test for a constant multiplier.

Test Three Terms Minimum: Never lock into a pattern rule after looking at only two terms; confirm it holds across at least three consecutive transitions.