# Can you write an equation to represent the nth term of the sequence?

To train for an upcoming marathon, Susie plans to run 2 miles per day the first week and then increase the daily distance by a half a mile each of the following weeks. How long before the marathon should she start?

Identify and label the variables.

Write an equation to represent the nth term of the sequence.

If the pattern continues during which week will she run 10 miles per day?

Is it reasonable to think that this pattern will continue indefinitely, explain?

How long before the marathon should she start?

Relevance
• 1 month ago

Let n = week number

d(n) = daily distance in week n

d(1) = 2

d(2) = d(1) + 0.5 = 2.5

d(3) = d(2) + 0.5 = 2.5 + 0.5 = 3

d(4) = d(3) + 0.5 = 3 + 0.5 = 3.5

d(5) = d(4) + 0.5 = 3.5 + 0.5 = 4

.

d(n) = d(n - 1) + 0.5

Looking at the pattern, the distance run has integer values when n = 1, 3, 5, … If you jot down some more terms of the sequence, or follow the pattern in your head, you'll find that she'll be running 10 miles a day in week 17.

The pattern can't continue indefinitely. That would mean that she would be running greater and greater distances each week, always. For example, she'll never be able to run 500 miles a day, which would be a part of the 'increasing indefinitely' sequence.

week … daily distance

1 ……….. 2

3 ……….. 3

5 ……….. 4

7 ……….. 5

.

.

17 ……… 10

That's a linear relationship. I won't type out all the steps, but you can find the equation of the line by using any of the two points (except the first one, as that one is sort of an initial condition, not really following the rule of the sequence.)

d = (1/2)x + (3/2)

If the marathon is 26 miles, want

26 = (1/2)x + (3/2)

52 = x + 3 ……………… multiplied both sides by 2 to get rid of the fractions

49 = x

In week 49, she would be running 26 miles a day. I suppose she should start this training 48 weeks before the marathon.