High School

Q1. Describe the rule for the patterns below:

a) 3; 8; 13

b) 100; 10; √10;

Q2) a) Work out the rule for the nth term of this pattern:
2; 4; 8; 16

b) Use your answer in 1 to work out the rule for the nth term of this pattern:
6; 12; 24; 48.

Answer :

Question 1:

a) 3; 8; 13

The sequence 3, 8, 13 is an arithmetic sequence where each term increases by a constant difference. Let's determine the rule for this sequence:

  1. Identify the common difference:

    • From 3 to 8: [tex]8 - 3 = 5[/tex]
    • From 8 to 13: [tex]13 - 8 = 5[/tex]

    The common difference is 5.

  2. Rule for the nth term:

    • The rule for an arithmetic sequence is given by the formula:
      [tex]a_n = a_1 + (n - 1) \times d[/tex]
      where [tex]a_1[/tex] is the first term and [tex]d[/tex] is the common difference.
    • Here, [tex]a_1 = 3[/tex] and [tex]d = 5[/tex].
    • So, the nth term rule is:
      [tex]a_n = 3 + (n - 1) \times 5[/tex]
      Simplifying gives:
      [tex]a_n = 3 + 5n - 5 = 5n - 2[/tex]

b) 100; 10;

[tex]\sqrt{10}[/tex]

This sequence does not have a constant difference, so it is not arithmetic. Let's see if it's geometric:

  1. Identify a possible pattern:
    • The transition seems to involve dividing by 10.
    • From 100 to 10: [tex]100 \div 10 = 10[/tex]
    • From 10 to [tex]\sqrt{10}[/tex]: it seems the sequence involves taking square roots or dividing further.

It appears to be geometric with a decreasing factor:

  • From 100 to 10, it divides by 10,
  • From 10 to [tex]\sqrt{10}[/tex], it takes the square root.

However, due to the lack of a clean pattern across all given terms, a strict rule cannot be determined without further context.

Question 2:

a) 2; 4; 8; 16

This sequence is geometric where each term is multiplied by 2:

  1. Identify the ratio:

    • From 2 to 4: [tex]4 \div 2 = 2[/tex]
    • From 4 to 8: [tex]8 \div 4 = 2[/tex]
    • From 8 to 16: [tex]16 \div 8 = 2[/tex]

  2. Rule for the nth term:

    • The rule for a geometric sequence is given by the formula:
      [tex]a_n = a_1 \times r^{n-1}[/tex]
      where [tex]a_1[/tex] is the first term and [tex]r[/tex] is the common ratio.
    • Here, [tex]a_1 = 2[/tex] and [tex]r = 2[/tex].
    • So, the nth term rule is:
      [tex]a_n = 2 \times 2^{n-1}[/tex]

b) 6; 12; 24; 48

This sequence follows a similar pattern to the previous one, but starting with 6:

  1. Identify the ratio:

    • Similar to the sequence in 2a, each term is doubled.

  2. Rule for the nth term:

    • Starting with [tex]a_1 = 6[/tex] and [tex]r = 2[/tex], the rule is:
      [tex]a_n = 6 \times 2^{n-1}[/tex]

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