Mason’s exercise plan lasts for six
minutes on the first day and increases by four minutes each day. For how long will Mason exercise on
the 18th day?
We can see that Mason increases his
exercise plan by the same amount of four minutes each day. This means that the times Mason
spends exercising daily form an arithmetic sequence with a common difference of
four. We’re also told that Mason spends
six minutes exercising on the first day of his plan, which means that the first term
of this arithmetic sequence 𝑎 is equal to six. We therefore have all the
information we need to write down as many terms of the sequence as we want or write
down the rule for the 𝑛th term.
The first term in this sequence is
six. The second term is four more than
this, so it’s 10. The third term is four more than
this, so it’s 14. We could continue in this way, but
it’s not very efficient if we need to get all the way to the 18th term in this
sequence. Instead, we can use the formula for
the 𝑛th term: 𝑎 sub 𝑛 is equal to 𝑎 plus 𝑛 minus one multiplied by 𝑑. Substituting six for 𝑎, the first
term, and four for 𝑑, the common difference, we have 𝑎 sub 𝑛 is equal to six plus
four multiplied by 𝑛 minus one.
We could simplify this
algebraically, or to find the 18th term, we could go straight to substituting 𝑛
equals 18. 𝑎 sub 18 is equal to six plus four
multiplied by 18 minus one. We have six plus four multiplied by
17. Four multiplied by 17 is 68. And adding six gives 74. Remember that the terms in the
sequence are times in minutes. So we found that the 18th term of
this sequence or the time Mason spends exercising on the 18th day is 74 minutes.