Nth-Term Rules and Sequence Types

Overview

These notes cover sequences defined by an nth-term formula, how to find terms, how to decide if a number is in a sequence, and how to find the nth-term rule from given terms, including special sequences (Fibonacci and geometric).

Using an nth-term rule to find terms

• The nth-term (position-to-term rule) gives the value of the term at position n in the sequence.
• To find specific terms, substitute n = 1, 2, 3, … into the nth-term expression.

Example: nth-term 5n + 1

• First term: n = 1 → 5(1) + 1 = 6
• Second term: n = 2 → 5(2) + 1 = 11
• Third term: n = 3 → 5(3) + 1 = 16
• 100th term: n = 100 → 5(100) + 1 = 501

Example: nth-term 9n − 3

• First term: n = 1 → 9(1) − 3 = 6
• Second term: n = 2 → 9(2) − 3 = 15
• Third term: n = 3 → 9(3) − 3 = 24

Example: nth-term 9n − 2 (first five terms)

• T₁: n = 1 → 9(1) − 2 = 7
• T₂: n = 2 → 9(2) − 2 = 16
• T₃: n = 3 → 9(3) − 2 = 25
• T₄: add 9 → 25 + 9 = 34
• T₅: add 9 → 34 + 9 = 43 (check: 9(5) − 2 = 43)

Summary table: evaluating nth-term rules

Testing whether a number is in a sequence

• Idea: If a number T is a term, there exists a whole number n such that nth-term expression = T.
• Solve the equation and check whether n is a whole number (positive integer).

Example: nth-term 3n − 5; is 95 in the sequence?

• Solve 3n − 5 = 95
  • Add 5: 3n = 100
  • Divide by 3: n = 100/3 = 33.\overline{3}
• n is not a whole number, so 95 is not in the sequence.
• Check nearby terms:
  • 33rd term: 3(33) − 5 = 99 − 5 = 94
  • 34th term: 3(34) − 5 = 102 − 5 = 97
  • 95 lies between 94 and 97, so it is not a term.

Example: nth-term 4n + 1; is 37 in the sequence?

• Solve 4n + 1 = 37
  • Subtract 1: 4n = 36
  • Divide by 4: n = 9
• n is a whole number, so 37 is a term; it is the 9th term.

Example: nth-term 9n − 2; is 80 in the sequence?

• Solve 9n − 2 = 80
  • Add 2: 9n = 82
  • Divide by 9: n = 82/9
• 82 is not in the 9 times table; n is not a whole number.
• 9 × 9 = 81, so 82/9 = 9 and 1/9.
• 80 is not in the sequence; it lies between the 9th and 10th terms.

Summary table: checking membership

Finding an nth-term rule from a sequence

General method for linear sequences (constant difference):

1. Find the term-to-term difference (d).
2. Write dn as the basic part (d times table).
3. Compare dn with the given sequence to find the constant adjustment (add or subtract a fixed number).
4. Write nth-term as dn ± constant.

Example: 7, 11, 15, 19, 23

• Differences: +4 each time → starts with 4n (4 times table).
• 4 times table: 4, 8, 12, 16, 20.
• Compare: sequence = times table + 3 each time.
• nth-term: 4n + 3.

Example: 4, 10, 16, 22, 28

• Differences: +6 each time → starts with 6n (6 times table).
• 6 times table: 6, 12, 18, 24, 30.
• Sequence is always 2 less than 6n.
• nth-term: 6n − 2.

Example: decreasing sequence with difference −5

• Sequence goes down by 5 each time → use −5n (negative 5 times table).
• −5 times table: −5, −10, −15, …
• To match the given sequence, we must add 29 to −5n.
• nth-term: −5n + 29 (or 29 − 5n).

Example: decreasing sequence with difference −3

• Sequence goes down by 3 each time → use −3n (negative 3 times table).
• −3 times table: −3, −6, −9, −12, −15.
• We must add 35 to −3n to get the given terms.
• nth-term: −3n + 35.

Example: 5, 12, 19, 26, 33

• Differences: +7 each time → start with 7n.
• 7 times table: 7, 14, 21, 28, 35.
• Sequence is 2 less than 7n each time.
• nth-term: 7n − 2.

Summary table: forming nth-term rules

Special sequences: Fibonacci and geometric

Fibonacci sequence

• Rule: each term (from the third onwards) is the sum of the previous two terms.
• Example given (first six terms shown; starts 1, 2, 3, 5, 8, 13 pattern implied by explanation):
  • 1st + 2nd → 3rd
  • 2nd + 3rd → 4th
  • 3rd + 4th → 5th
  • 4th + 5th → 6th
• Finding next terms:
  • 5th term = 3, 6th term = 5 → 7th term = 3 + 5 = 8
  • 8th term = 5 + 8 = 13
• Could continue: next would be 8 + 13 = 21, then 13 + 21, etc.

Geometric sequence (multiplicative pattern)

• Rule: multiply by the same number each time to get the next term.
• Example: doubling sequence
  • Each term is 2 × previous term.
  • Next term after 48: 2 × 48 = 96
  • Next term after 96: 2 × 96 = 192

Summary table: special sequences

Key Terms & Definitions

• nth-term: An expression that gives the term at position n in a sequence.
• Position-to-term rule: Another name for the nth-term rule.
• Term-to-term rule: The rule that describes how to get from one term to the next (e.g. “add 4”, “multiply by 2”).
• Linear sequence: A sequence with a constant difference between terms (e.g. +4 each time).
• Geometric sequence: A sequence where each term is obtained by multiplying the previous term by a fixed number.
• Fibonacci sequence: A sequence where each term is the sum of the previous two terms.
• Whole number (for n): Positive integer index (1, 2, 3, …); required for a value to be a valid term position.

Action Items / Next Steps

• Practise finding first several terms for given nth-term formulas by substitution.
• Practise testing whether a given number is in a sequence by solving an equation for n.
• Practise deriving nth-term rules: find the difference, form dn, then determine and apply the constant adjustment.
• Learn to recognise Fibonacci and geometric sequences from their defining patterns.