Question

Samantha is training for a race. The distances of her training runs form an arithmetic sequence. She runs 1 mi the first day and 2 mi the seventh day.

a. What is the explicit definition for this sequence?



b. How far does she run on day 19?

Answer 1

The explicit definition for this arithmetic sequence is an = a1 + (n - 1)d, where an is the nth term, a1 is the first term, n is the position of the term, and d is the common difference. Samantha runs 4 miles on day 19.

Step-by-step explanation:

a. The explicit definition for this arithmetic sequence can be found by using the formula:
an = a1 + (n - 1)d
Where an represents the nth term of the sequence, a1 is the first term, n is the position of the term, and d is the common difference between terms. In this case, the first term is 1 and the seventh term is 2.
a7 = 1 + (7 - 1)d
2 = 1 + 6d
6d = 1
d = 1/6

So, the arithmetic sequence can be written as: 1, 1 + d, 1 + 2d, 1 + 3d, ... 1 + 6d

b. To find the distance Samantha runs on day 19, we can use the same formula:
an = a1 + (n - 1)d
Since the first term is 1 and d is 1/6, we have:
a19 = 1 + (19 - 1)(1/6)
= 1 + 18/6
= 1 + 3
= 4 miles