The dots below are arranged in a single row that grows from one pattern to the next, always adding the dots on the right-hand end; the first three patterns ...
Question 1 Report
The dots below are arranged in a single row that grows from one pattern to the next, always adding the dots on the right-hand end; the first three patterns are shown.
- Write down the number of dots in Pattern 4. (1)
- Find, in terms of \(n\), an expression for the dots in Pattern \(n\). (2)
- Work out the pattern number that has exactly 101 dots. (2)
Answer Details
An arithmetic sequence with a formula \(a_1 + (n-1)d\) uses the first term \(a_1\) and the common difference \(d\); once the pattern's growth is known, both the next pattern and any given pattern number can be found without drawing every step.
The patterns shown have 6, 11 and 16 dots, each 5 more than the last, so Pattern 4 continues by adding 5 more dots: \[16 + 5 = 21 \text{ dots}\] [1 mark]
The common difference is \[d = 5\] Using the first term 6 and this common difference, the \(n\)th term is \(6+(n-1)(5)\), which expands to \[5n + 1\] [2 marks]
Setting \(5n+1=101\) and rearranging gives \(5n=100\), so \[n = 20\] [2 marks]
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