Under one phone tariff the monthly bills, in pounds, form the sequence 18, 22, 26, 30, and so on. Write down the common difference of the sequence. (1) Find...
Question 1 Report
Under one phone tariff the monthly bills, in pounds, form the sequence 18, 22, 26, 30, and so on.
- Write down the common difference of the sequence. (1)
- Find an expression for the \(n\)th term. (2)
Answer Details
The bills go up by the same amount each month, so this is an arithmetic sequence and the \(n\)th term is linear in \(n\).
(a) Subtracting consecutive terms: \(22 - 18 = 4\), \(26 - 22 = 4\), \(30 - 26 = 4\). The common difference is \(4\). [1]
(b) Because the difference is constant at \(4\), the \(n\)th term has the form \(4n + c\). The \(4n\) part alone would give \(4, 8, 12, 16, \ldots\), and each actual term is \(14\) more than that. Finding \(c\) formally from the first term:
\[4 \times 1 + c = 18 \implies c = 14\][1]
\[n\text{th term} = 4n + 14\][1]
Test it on a term that was not used to build it: \(4 \times 3 + 14 = 26\), which matches the third bill. The coefficient of \(n\) is always the common difference, and the constant is the value the pattern would have at \(n = 0\), here \(18 - 4 = 14\). Writing \(4n + 18\) is the standard error and gives \(22\) rather than \(18\) for the first month.
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