Video: Finding the Sum of the Terms of an Arithmetic Sequence given Its First Term and the Common Difference in a Word Problem

Chloe devised a week-long study plan to prepare for finals. On the first day she plans to study for 1 hour, and she will increase her study time by 30 minutes each successive day. How many total hours will Chloe have spent studying at the end of the week?

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Video Transcript

Chloe devised a week-long study plan to prepare for finals. On the first day, she plans to study for one hour and she will increase her study time by 30 minutes each successive day. How many total hours will Chloe have spent studying at the end of the week?

We can model this problem using an arithmetic series as Chloe is increasing her study time by the same amount each day. We can let 𝑎 be the first term, 𝑑 is the common difference, and 𝑛 is the number of terms. We can then use the formula 𝑆 of 𝑛 is equal to 𝑛 divided by two multiplied by two 𝑎 plus 𝑛 minus one multiplied by 𝑑 to work out the total amount of time that Chloe will spend studying.

𝑎 will be equal to one as on the first day Chloe studied for one hour. 𝑑 is equal to 0.5 as she is increasing her study time by 30 minutes each day. 30 minutes is equal to half an hour. 𝑛 is equal to seven as there are seven days in a week and we want to calculate the total number of hours that Chloe will have spent studying at the end of the week.

Substituting these numbers into our formula gives us seven divided by two multiplied by two multiplied by one plus seven minus one multiplied by 0.5. Seven divided by two is 3.5, two multiplied by one is two, and seven minus one which is six multiplied by 0.5 is equal to three. Two plus three is equal to five. Therefore, we are left with 3.5 multiplied by five. This is equal to 17.5. Therefore, Chloe will have studied for a total of 17.5 hours if on the first day she studies for one hour and increases her study time by 30 minutes on each successive day.

We could also have solved this problem by working out how long Chloe studies full on each of the seven days. On the first day, she studied for one hour. As she increases her study time by 30 minutes each day, on the second day she would study for one and a half hours or 1.5 hours.

Continuing this pattern shows us that Chloe would study for two hours on day three, two and a half hours on day four, three hours on day five, three and a half hours on day six, and four hours on day seven. Adding these seven numbers gives us a total of 17.5 hours.

Chloe’s total study time is 17 and a half hours.

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