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Number - Number Sequences and Pattern Rules

Grade 8IB

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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A number sequence is a list of numbers following a specific rule. Each number in the sequence is called a 'term', represented as unu_n, where nn is the position of the term.

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An Arithmetic Sequence (or Linear Sequence) is one where the difference between consecutive terms is constant. This constant is called the common difference (dd).

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The nthn^{th} term (unu_n) is a general rule or formula that allows you to calculate any term in a sequence based on its position nn.

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A Geometric Sequence is one where each term is found by multiplying the previous term by a fixed, non-zero number called the common ratio (rr).

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To find the rule for a linear sequence, identify the common difference (dd) and the first term (aa). The rule is generally in the form un=dn+cu_n = dn + c.

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Term-to-term rules describe how to get from one term to the next (e.g., 'add 55'), while position-to-term rules (the nthn^{th} term) allow you to jump to any position.

📐Formulae

un=a+(n−1)du_n = a + (n - 1)d

d=un−un−1d = u_n - u_{n-1}

un=a×r(n−1)u_n = a \times r^{(n-1)}

r=unun−1r = \frac{u_{n}}{u_{n-1}}

💡Examples

Problem 1:

Find the nthn^{th} term rule for the sequence: 5,8,11,14,…5, 8, 11, 14, \dots

Solution:

un=3n+2u_n = 3n + 2

Explanation:

  1. Find the common difference: 8−5=38 - 5 = 3 and 11−8=311 - 8 = 3. So, d=3d = 3.
  2. The formula starts with 3n3n.
  3. When n=1n = 1, 3(1)=33(1) = 3. To get the first term 55, we need to add 22.
  4. Therefore, un=3n+2u_n = 3n + 2. Testing for n=2n=2: 3(2)+2=83(2) + 2 = 8, which is correct.

Problem 2:

Given the geometric sequence 2,6,18,54,…2, 6, 18, 54, \dots, find the 7th7^{th} term.

Solution:

u7=1458u_7 = 1458

Explanation:

  1. Identify the first term a=2a = 2.
  2. Find the common ratio r=62=3r = \frac{6}{2} = 3.
  3. Use the formula un=a×r(n−1)u_n = a \times r^{(n-1)}.
  4. For n=7n = 7: u7=2×3(7−1)=2×36u_7 = 2 \times 3^{(7-1)} = 2 \times 3^6.
  5. 36=7293^6 = 729.
  6. u7=2×729=1458u_7 = 2 \times 729 = 1458.

Problem 3:

Determine if 102102 is a term in the sequence un=4n−3u_n = 4n - 3.

Solution:

102102 is not a term because nn is not an integer.

Explanation:

  1. Set the formula equal to the target number: 4n−3=1024n - 3 = 102.
  2. Solve for nn: 4n=1054n = 105.
  3. n=1054=26.25n = \frac{105}{4} = 26.25.
  4. Since nn must be a whole number (a position in the sequence), 102102 is not a term in this sequence.