Mathematical Sequences and Position Rules

Rules for Arithmetic Sequences Based on Position

• To find unknown terms in a sequence where the terms increase by a constant amount, a general rule can be applied based on the position of the term in the sequence (first position, second position, etc.).

• The fundamental rule described is to multiply the position number by the first number in the sequence (the common difference).

• Sequence A: Multiples of 3

  • Terms: $3, 6, 9, 12, 15, 18$
  • Rule: Multiply the position number by $3$ ($3 \times \text{position}$).
  • Multiplication Sentence Example: To find the value for the $6^{th}$ position, use $3 \times 6 = 18$.
  • Unknown Terms Identified: The last three terms are $12$, $15$, and $18$.

• Sequence B: Multiples of 5

  • Terms: $5, 10, 15, 20, 25, 30, 35$
  • Rule: Multiply the position number by $5$ ($5 \times \text{position}$).
  • Multiplication Sentence Example: To find the value for the $7^{th}$ position, use $5 \times 7 = 35$.
  • Unknown Terms Identified: The sequence completes with $20, 25, 30, 35$.

• Sequence C: Multiples of 6

  • Terms: $6, 12, 18, 24, 30, 36, 42, 48$
  • Rule: Multiply the position number by $6$ ($6 \times \text{position}$).
  • Multiplication Sentence Example: To find the value for the $8^{th}$ position, use $6 \times 8 = 48$.
  • Unknown Terms Identified: These terms follow the first three: $24, 30, 36, 42, 48$.

• Sequence D: Multiples of 9

  • Terms: $9, 18, 27, 36, 45, 54, 63, 72, 81$
  • Rule: Multiply the position number by $9$ ($9 \times \text{position}$).
  • Multiplication Sentence Example: To find the value for the $9^{th}$ position, use $9 \times 9 = 81$.
  • Unknown Terms Identified: The subsequent terms in the sequence are $36, 45, 54, 63, 72, 81$.

Calculating Specific nth Terms

• Determining a distant term in a sequence involves identifying the underlying pattern and applying it to the requested position number.

• Sequence: Multiples of 2

  • Sequence Start: $2, 4, 6, ...$
  • Calculation for the $12^{th}$ Term: $2 \times 12 = 24$
  • Answer: The $12^{th}$ term is $24$.

• Sequence: Multiples of 10

  • Sequence Start: $10, 20, 30, ...$
  • Calculation for the $15^{th}$ Term: $10 \times 15 = 150$
  • Answer: The $15^{th}$ term is $150$.

• Sequence: Multiples of 7

  • Sequence Start: $7, 14, 21, ...$
  • Calculation for the $9^{th}$ Term: $7 \times 9 = 63$
  • Answer: The $9^{th}$ term is $63$.

Tabular Analysis of Number Sequences

• Tables are a structured way to map the relationship between a position ($n$) and its corresponding term ($T$).

• Table A: Term = $5 \times \text{Position}$

  • Position 1: Term is $1 \times 5 = 5$
  • Position 2: Term is $2 \times 5 = 10$
  • Position 3: Term is $3 \times 5 = 15$
  • Position 11: Term is $11 \times 5 = 55$
  • Position 20: Term is $20 \times 5 = 100$
  • Position 29: Term is $29 \times 5 = 145$
  • Summary of Results: The calculated terms for positions $11, 20,$ and $29$ are $55, 100,$ and $145$ respectively (with $20$ as a specific position value identified in the summary).

• Table B: Term = $8 \times \text{Position}$

  • Position 1: Term is $1 \times 8 = 8$
  • Position 2: Term is $2 \times 8 = 16$
  • Position 3: Term is $3 \times 8 = 24$
  • Position 6: Term is $6 \times 8 = 48$
  • Position 12: Term is $12 \times 8 = 96$
  • Position 25: Term is $25 \times 8 = 200$
  • Summary of Results: The calculated terms for positions $6, 12,$ and $25$ identify the sequence values $48, 96,$ and $200$ respectively.

Square Numbers and Their Positional Relationships

• A square number is the result of multiplying a number by itself. This can be expressed as the position number multiplied by the position number ($\text{position} \times \text{position}$).

• Determining Square Numbers by Position:

  • Fourth Position: $4 \times 4 = 16$
  • Sixth Position: $6 \times 6 = 36$
  • Seventh Position: $7 \times 7 = 49$
  • Tenth Position: $10 \times 10 = 100$

• Determining Position Based on the Square Number:

  • Square Number 25: Because $5 \times 5 = 25$, it is in the $5^{th}$ position.
  • Square Number 81: Because $9 \times 9 = 81$, it is in the $9^{th}$ position.
  • Square Number 49: Because $7 \times 7 = 49$, it is in the $7^{th}$ position.
  • Square Number 64: Because $8 \times 8 = 64$, it is in the $8^{th}$ position.