3.2 Sequences, Patterns & Series Calculations

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 $d$ ($a_n = a_1 + (n-1)d$). In geometric sequences, terms change by a constant multiplier ratio $r$ ($a_n = a_1 \cdot r^{n-1}$). To quickly sum an arithmetic series, use $S_n = \frac{n}{2}(a_1 + a_n)$.

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.

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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 ($d$).

Key Arithmetic Formulas

$\text{Common Difference: } d = a_{k+1} - a_k$ $\text{General Term Formula: } a_n = a_1 + (n - 1)d$

Where:

• $a_n$ = the value of the $n$-th term
• $a_1$ = the first term in the sequence
• $n$ = the position number of the target term
• $d$ = the common difference

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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

• $a_1 = 45$
• $a_2 = 58$
• $a_3 = 71$
• Common difference $d = 58 - 45 = 13$ (and $71 - 58 = 13$)
• Target term position $n = 18$

Step 2: Apply the General Term Formula

$a_{18} = a_1 + (18 - 1)d$ $a_{18} = 45 + (17 \times 13)$

Step 3: Compute the Multiplication and Final Sum

$17 \times 13 = 221$ $a_{18} = 45 + 221 = 266 \text{ crates}$

Shipment 18 will deliver 266 crates.

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Arithmetic Series Summation ($S_n$)

When an AFQT question asks for the total accumulation or total sum of an arithmetic sequence over $n$ steps, adding every individual term manually wastes valuable test time and introduces calculation errors. Use Gauss's Summation Formula:

$S_n = \frac{n}{2} (a_1 + a_n) \quad \text{or} \quad S_n = \frac{n}{2} [2a_1 + (n - 1)d]$

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

• $a_1 = 12$
• $d = 3$
• $n = 30$

Step 2: Determine the Final Day's Value ($a_{30}$)

$a_{30} = a_1 + (30 - 1)d = 12 + (29 \times 3) = 12 + 87 = 99 \text{ push-ups}$

Step 3: Calculate the Series Sum ($S_{30}$)

$S_{30} = \frac{30}{2} (a_1 + a_{30})$ $S_{30} = 15 \times (12 + 99)$ $S_{30} = 15 \times 111 = 1,665 \text{ push-ups}$

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

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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 ($r$).

Key Geometric Formulas

$\text{Common Ratio: } r = \frac{a_{k+1}}{a_k}$ $\text{General Term Formula: } a_n = a_1 \cdot r^{n-1}$

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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

• $a_1 = 4$
• $a_2 = 12$
• Common ratio $r = \frac{12}{4} = 3$ (verifying: $\frac{36}{12} = 3$)
• Target hour $n = 6$

Step 2: Substitute into Geometric Formula

$a_6 = a_1 \cdot r^{6-1}$ $a_6 = 4 \cdot 3^5$

Step 3: Evaluate Exponent and Multiply

$3^5 = 3 \times 3 \times 3 \times 3 \times 3 = 243$ $a_6 = 4 \times 243 = 972 \text{ servers}$

At Hour 6, exactly 972 servers will be infected.

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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:

$\begin{array}{lccccccccccc} \text{Given Terms:} & 3 & & 7 & & 13 & & 21 & & 31 & & 43 \\ \text{1st Differences:} & & +4 & & +6 & & +8 & & +10 & & +12 & \\ \text{2nd Differences:} & & & +2 & & +2 & & +2 & & +2 & & \end{array}$

Finding the Next Term

1. The 1st differences ($4, 6, 8, 10, 12$) increase by $+2$ each step.
2. The next 1st difference must be $12 + 2 = 14$.
3. Add this difference to the last term: $43 + 14 = 57$.

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Interleaved and Alternating Sequences

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

Example of Interleaved Pattern

$\mathbf{5}, 100, \mathbf{10}, 95, \mathbf{15}, 90, \mathbf{20}, \underline{?}$

Analysis: Split into Odd and Even Indices

• Odd Terms (Positions 1, 3, 5, 7): $5, 10, 15, 20 \dots$ (Increases by $+5$)
• Even Terms (Positions 2, 4, 6, 8): $100, 95, 90, \underline{?}$ (Decreases by $-5$)

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

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Sequence Formula Reference Summary

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AFQT Exam Strategy & Rapid Pattern Recognition

1. First Check Differences: Always test for a constant first difference before assuming a complex rule.
2. Second Check Ratios: If differences expand rapidly ($2, 6, 18, 54$), test for a constant multiplier.
3. 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.