3.3 Number Sequences & Pattern Recognition

3.3 Number Sequences & Pattern Recognition

Quick Summary: A number sequence is an ordered list of numbers following a systematic mathematical rule. The EIAT heavily tests pattern recognition through arithmetic sequences (constant common difference $d$), geometric sequences (constant common ratio $r$), multi-operational rules, and alternating progressions. Electricians utilize sequence formulas ($a_n = a_1 + (n-1)d$ and $a_n = a_1 \cdot r^{n-1}$) to solve practical problems involving transformer voltage step-up/step-down ratios, multi-stage motor starting currents, and ladder resistance networks.

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Fundamentals of Number Sequences

A sequence is an ordered set of numbers where each individual number is called a term. Terms are designated by subscripted variables:

$$a_1, a_2, a_3, a_4, \dots, a_n$$

• $a_1$ represents the first term of the sequence.
• $a_n$ represents the $n$-th term (also called the general term or explicit formula).
• $n$ represents the term position index ($n = 1, 2, 3, \dots$).

Part 1 of the EIAT features numerous number series items. Recognizing sequence structures allows candidates to find missing intermediate terms or extrapolate distant terms efficiently.

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Arithmetic Sequences ($a_n = a_1 + (n-1)d$)

An arithmetic sequence (or arithmetic progression) is a sequence in which the difference between any two consecutive terms is constant.

The Common Difference ($d$)

The constant change between consecutive terms is called the common difference ($d$):

$$d = a_{k+1} - a_k$$

• If $d > 0$, the sequence is increasing (e.g., $5, 11, 17, 23, \dots \implies d = +6$).
• If $d < 0$, the sequence is decreasing (e.g., $100, 85, 70, 55, \dots \implies d = -15$).

Explicit Formula for the $n$-th Term

To find any term $a_n$ without writing out all preceding terms, use the explicit arithmetic formula:

$$a_n = a_1 + (n - 1)d$$

Derivation Logic

• $a_1 = a_1$
• $a_2 = a_1 + d$
• $a_3 = a_1 + 2d$
• $a_4 = a_1 + 3d$
• $\dots$
• To reach the $n$-th term, you must add the common difference $d$ exactly $(n - 1)$ times to the initial term $a_1$.

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Step-by-Step Example: Finding a Large Term

Problem: What is the 30th term of the arithmetic sequence: $11, 17, 23, 29, \dots$?

1. Identify the first term ($a_1$): $$a_1 = 11$$
2. Calculate the common difference ($d$): $$d = 17 - 11 = 6 \quad (\text{Verify: } 23 - 17 = 6, \ 29 - 23 = 6)$$
3. Identify the target position ($n$): $$n = 30$$
4. Apply the explicit formula: $$a_{30} = a_1 + (30 - 1)d$$ $$a_{30} = 11 + (29) \times 6$$
5. Perform multiplication before addition: $$29 \times 6 = (30 - 1) \times 6 = 180 - 6 = 174$$ $$a_{30} = 11 + 174 = 185$$

Thus, the 30th term of the sequence is 185.

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Geometric Sequences ($a_n = a_1 \cdot r^{n-1}$)

A geometric sequence (or geometric progression) is a sequence in which each term after the first is obtained by multiplying the previous term by a constant non-zero number called the common ratio ($r$).

The Common Ratio ($r$)

$$r = \frac{a_{k+1}}{a_k}$$

• If $r > 1$, the sequence exhibits exponential growth (e.g., $3, 6, 12, 24, 48, \dots \implies r = 2$).
• If $0 < r < 1$, the sequence exhibits exponential decay (e.g., $1000, 500, 250, 125, \dots \implies r = 0.5$).
• If $r < 0$, the sequence terms alternate in sign (e.g., $5, -15, 45, -135, \dots \implies r = -3$).

Explicit Formula for the $n$-th Term

The general term of a geometric sequence is given by:

$$a_n = a_1 \cdot r^{n-1}$$

Example

To find the 7th term of $2, 6, 18, 54, \dots$:

• $a_1 = 2$, $r = \frac{6}{2} = 3$, $n = 7$.
• $a_7 = 2 \cdot 3^{7-1} = 2 \cdot 3^6 = 2 \cdot 729 = 1,458$.

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Advanced EIAT Pattern Recognition Strategies

EIAT number series questions often extend beyond simple arithmetic or geometric progressions. Test takers must recognize multi-operational, polynomial, and alternating patterns.

1. Two-Step Operational Sequences ($a_n = k \cdot a_{n-1} + c$)

In these patterns, obtaining the next term requires combining two operations—such as multiplying by a constant and adding/subtracting a number.

Step-by-Step Example: Missing Term Identification

Problem: What is the missing number in the sequence: $7, 15, 31, 63, \underline{\quad}, 255$?

1. Analyze term differences:

   • $15 - 7 = 8$
   • $31 - 15 = 16$
   • $63 - 31 = 32$ Notice that the differences double each step ($8, 16, 32, \dots$).

2. Formulate the recursive rule:

   • Test multiplication and addition: $$7 \times 2 + 1 = 14 + 1 = 15$$ $$15 \times 2 + 1 = 30 + 1 = 31$$ $$31 \times 2 + 1 = 62 + 1 = 63$$

   The operational pattern is: Multiply the previous term by 2 and add 1.

3. Calculate the missing term ($a_5$):

   $$a_5 = 63 \times 2 + 1 = 126 + 1 = 127$$

4. Verify with the subsequent term ($a_6 = 255$):

   $$127 \times 2 + 1 = 254 + 1 = 255 \quad \checkmark$$

The missing number is 127.

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2. Perfect Square & Cube Sequences

Check if the sequence terms (or the differences between terms) correspond to squares ($n^2$) or cubes ($n^3$):

• Pure Squares: $1, 4, 9, 16, 25, 36, 49, \dots$
• Square Differences: $2, 3, 7, 16, 32, 57, \dots$ (Differences are $1^2, 2^2, 3^2, 4^2, 5^2 \implies +1, +4, +9, +16, +25$).

3. Interleaved (Alternating) Sequences

Some test questions combine two independent sequences into a single alternating series:

• Example: $5, 50, 10, 45, 15, 40, \underline{\quad}, \underline{\quad}$
• Odd Position Terms ($a_1, a_3, a_5, a_7$): $5, 10, 15, \mathbf{20}$ ($+5$ progression).
• Even Position Terms ($a_2, a_4, a_6, a_8$): $50, 45, 40, \mathbf{35}$ ($-5$ progression).

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Trade Applications: Motor Currents & Transformer Taps

Pattern recognition and sequence formulas directly translate to industrial electrical installations.

Application 1: Reduced-Voltage Motor Start-Up Current Progression

Heavy industrial electric motors draw severe inrush currents during direct-on-line starting. To prevent utility voltage sags, electricians install multi-stage reduced-voltage soft starters or autotransformer starters that step down starting current across sequential timing stages.

An electrician logs motor current draws across 5 starting stages:

• Stage 1: $321\text{ A}$
• Stage 2: $281\text{ A}$
• Stage 3: $241\text{ A}$
• Stage 4: $201\text{ A}$

Analysis

1. Calculate the common difference ($d$): $$d = 281 - 321 = -40\text{ A}$$ $$241 - 281 = -40\text{ A}$$ $$201 - 241 = -40\text{ A}$$
2. Recognize an arithmetic progression with initial term $a_1 = 321\text{ A}$ and common difference $d = -40\text{ A}$.
3. Predict the expected Stage 5 current ($a_5$): $$a_5 = a_4 + d = 201 - 40 = 161\text{ A}$$ (Using formula: $a_5 = 321 + (5-1)(-40) = 321 - 160 = 161\text{ A}$)

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Application 2: Multi-Tap Step-Down Transformers

Distribution transformers frequently utilize primary or secondary tap sequences configured in geometric ratios (e.g., $\pm 2.5\%$ and $\pm 5.0\%$ voltage tap steps). Calculating step voltages across multi-tap windings relies on geometric ratio multiplication, allowing electricians to set tap changers accurately under varying line conditions.