Question Video: Solving Word Problems Involving Sequences Mathematics

Video Transcript

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.