A workshop schedules a customer's service intervals to increase by a fixed number of miles each time, to match the car's improving reliability. The lengths ...
Question 1 Report
A workshop schedules a customer's service intervals to increase by a fixed number of miles each time, to match the car's improving reliability. The lengths of the 7th and 8th intervals are shown on the diagram.
- Write down the increase in interval length from one service to the next, in miles. (2)
- Work out the length of the 1st interval, in miles. (2)
Answer Details
This question tests reading two consecutive terms of an arithmetic sequence from a number-line diagram and using them to find the first term.
(a) The diagram shows the 7th interval as \(6000\) miles and the 8th interval as \(6500\) miles. The increase from one service interval to the next is:
\[6500-6000 = 500 \text{ miles}\][2]
(b) Let \(u_1\) be the length of the 1st interval. The 8th interval is \(7\) increases after the 1st, so:
\[u_1 + 500(7) = 6500\] \[u_1 = 6500-3500 = 3000\][2]
The 1st interval is \(3000\) miles. Check: \(3000+500(7)=3000+3500=6500\), matching the given 8th interval, and \(3000+500(6)=6000\), matching the given 7th interval.
When two consecutive terms of a sequence are given directly (rather than the first term), the common difference is found immediately by subtraction, and the first term is then found by counting backwards the correct number of steps, here \(7\) steps back from the 8th interval to the 1st.
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