To help get in shape, Joe gets a new pair of running shoes. If Joe runs 1 mi each day in week 1

To help get in shape, Joe gets a new pair of running shoes....

Key Concepts

Arithmetic Series

An arithmetic series is the sum of the terms of an arithmetic sequence. Knowing how to calculate this sum using formulas—which typically involve the first term, the number of terms, and the common difference—enables efficient computation of the total effect of consistent incremental changes.

Arithmetic Sequence

An arithmetic sequence is a series of numbers where each term is obtained by adding a constant difference to the previous term. This concept is crucial when modeling situations with regular, incremental change over time.

Transcript

00:01 All right, for problem 37, we get a problem about a runner, and he wants to run one mile per day for the first week, all seven days.

00:15 For the first week, he'll run one mile each day.

00:18 And then the second week, he wants to add a tenth of a mile to his routine each day.

00:23 So he'll run one and a tenth for all seven days, the second week.

00:27 And then for the third week he'll run one and two tenths of a mile every day.

00:35 So he's going to add a tenth of a mile each week, and he wants to know how much he's going to run total after 25 weeks, the total mileage.

00:46 So luckily we can use a summation to help us figure this out quicker than just adding this up week by week.

00:54 And so we're thinking that this runner, we're just going to figure this out weekly.

01:06 We're going to multiply.

01:07 We're going to go from week one to week 25.

01:14 And we're thinking he's going to start with just one mile.

01:18 So we would, you know, that would just be 20, 25 miles.

01:23 Of course, there's seven days in a week.

01:25 So we're going to have to multiply all this by seven.

01:30 So if he ran one mile the first week and one mile the second week and one mile the third week, that'd be a total of 25 miles times seven, seven days in the week.

01:42 But it's not that simple.

01:44 He's going to add one -tenth of a mile each week.

01:56 If i put an eye here, because there has to be an eye in here in this expression somewhere because we have to plug in a one and plug in a two and plug in a three.

02:07 If i just leave it as an eye and i plug a one in, that would mean, the first week he's going to run one plus one tenth times one, which would be one and a tenth the first week.

02:22 But that's not true.

02:23 He's only running one mile the first week.

02:28 And then if i plug in a two, he'd run one in two tenths the second week, which is, so we're a little bit ahead of ourselves.

02:37 So we actually have to subtract one from this to get it to line up with the proper weeks.

02:48 So now if i plug a 1 in for i, 1 minus 1 is 0, 0 times a 10th times a 10th is 0 plus 1.

02:56 So the first week, he's running one mile.

03:00 Then if i plug in a 2 to represent the second week, 2 minus 1 is 1, 1 times a 10th is a 10th, and a 10th plus 1 is 1 in a 10th the second week, which is exactly what we wanted.

03:13 And then we could carry that on for the third week.

03:16 Could be one in two tenths, and for the fourth week it could be one in three tenths...

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