Patterns & Sequences • Topic 2 of 3

Arithmetic Progressions (AP)

What are Geometric Patterns?

A geometric pattern (or geometric sequence) is a sequence where each term is found by multiplying the previous term by a constant. This constant is called the common ratio (\(r\)).

Formula for Geometric Sequence:

\(a_n = a_1 \times r^{(n-1)}\)

Where:

  • \(a_n\) = the \(n\)th term
  • \(a_1\) = first term
  • \(r\) = common ratio
  • \(n\) = position of the term

Examples of Geometric Sequences:

SequenceFirst Term (\(a_1\))Common Ratio (\(r\))Next Term
2, 6, 18, 54, ...2×3162
100, 50, 25, 12.5, ...100×0.5 or ÷26.25
4, -12, 36, -108, ...4×(-3)324
5, 5, 5, 5, ...5×15

What are Recursive Patterns?

A recursive pattern defines each term based on the previous term(s) using a fixed rule. Instead of giving a direct formula (like \(a_n = 2n+3\)), recursive patterns tell you how to get from one term to the next.

Recursive Formula Format:

  • \(a_1 =\) [first term]
  • \(a_n =\) [rule involving \(a_{n-1}\)]

Examples of Recursive Patterns:

TypeRecursive RuleFirst TermGenerated Sequence
Arithmetic\(a_n = a_{n-1} + 5\)\(a_1 = 2\)2, 7, 12, 17, ...
Geometric\(a_n = 3 \times a_{n-1}\)\(a_1 = 4\)4, 12, 36, 108, ...
Fibonacci\(a_n = a_{n-1} + a_{n-2}\)\(a_1 = 1, a_2 = 1\)1, 1, 2, 3, 5, 8, ...

Comparison: Arithmetic vs Geometric Sequences

FeatureArithmetic SequenceGeometric Sequence
OperationAddition/SubtractionMultiplication/Division
Common ElementDifference (\(d\))Ratio (\(r\))
Formula\(a_n = a_1 + (n-1)d\)\(a_n = a_1 \times r^{(n-1)}\)
Graph ShapeStraight lineCurve (exponential)
Arithmetic Progression (AP)3a1+58a2+513a3+518a4+523a5+528a6Arithmetic Progression Formulasnᵗʰ Term: aₙ = a + (n−1)dSum of n terms: Sₙ = n/2 × [2a + (n−1)d]or Sₙ = n/2 × (first term + last term)
Solved Examples
1
Worked Example
Example 1: Determine if the sequence 5, 15, 45, 135 is geometric. If yes, find the common ratio and the 8th term.
Solution- Step 1: Check ratios between consecutive terms - Step 2: \(15 ÷ 5 = 3\) - Step 3: \(45 ÷ 15 = 3\) - Step 4: \(135 ÷ 45 = 3\) - Step 5: All ratios equal 3, so it is geometric with \(r = 3\) - Step 6: \(a_1 = 5\) - Step 7: Use formula \(a_n = a_1 \times r^{(n-1)}\) - Step 8: \(a_8 = 5 \times 3^{(8-1)} = 5 \times 3^7\) - Step 9: \(3^7 = 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 = 2187\) - Step 10: \(5 \times 2187 = 10935\)
2
Worked Example
Example 2: Write the recursive formula for the sequence: 7, 10, 13, 16, 19, ...
Solution- Step 1: Identify the pattern: each term increases by 3 - Step 2: \(a_1 = 7\) (first term) - Step 3: For \(n \geq 2\), \(a_n = a_{n-1} + 3\) - Step 4: Write recursive definition: - \(a_1 = 7\) - \(a_n = a_{n-1} + 3\) for \(n \geq 2\)
3
Worked Example
Example 3: A ball is dropped from a height of 64 metres. Each time it bounces, it rises to half the previous height. Write the first 5 terms of the sequence of bounce heights. Is this arithmetic or geometric? Find the height after the 6th bounce.
Solution- Step 1: Initial height (first drop) = 64 m (this is \(a_1\)) - Step 2: After first bounce, height = \(64 \times \frac{1}{2} = 32\) m (\(a_2\)) - Step 3: After second bounce, height = \(32 \times \frac{1}{2} = 16\) m (\(a_3\)) - Step 4: After third bounce, height = \(16 \times \frac{1}{2} = 8\) m (\(a_4\)) - Step 5: After fourth bounce, height = \(8 \times \frac{1}{2} = 4\) m (\(a_5\)) - Step 6: Sequence: 64, 32, 16, 8, 4, ... - Step 7: Each term is multiplied by \(\frac{1}{2}\), so it is geometric with \(r = \frac{1}{2}\) - Step 8: Height after 6th bounce = \(a_7\) (since \(a_1\) is initial, \(a_2\) is after 1st bounce) - Step 9: \(a_7 = a_1 \times r^{(7-1)} = 64 \times (\frac{1}{2})^6 = 64 \times \frac{1}{64} = 1\)

Key Points

  • Geometric sequence: Multiply each term by constant \(r\) (common ratio)
  • Common ratio: \(r = a_2 \div a_1 = a_3 \div a_2 = ...\)
  • Geometric formula: \(a_n = a_1 \times r^{(n-1)}\)
  • Recursive pattern: Defines each term using previous term(s)
  • Recursive has two parts: first term + rule to get next term
  • Arithmetic = addition/subtraction | Geometric = multiplication/division
  • Fibonacci is a famous recursive pattern (\(a_n = a_{n-1} + a_{n-2}\))
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Quick Check — 4 MCQs
Tap an option to check your answer0 / 4
Q1.In an AP, the difference between consecutive terms is:
Explanation: Common difference $d$.
Q2.The $n$-th term of an AP is:
Explanation: $a+(n-1)d$.
Q3.For $2, 5, 8, 11$, the common difference is:
Explanation: $d=3$.
Q4.The next term of $3, 7, 11, \ldots$ is:
Explanation: Add $4$.
Practice more MCQs → 📝 Assignment · 35 marks