Question

Shelly is training for a triathlon. In the triathlon she will ride her bike run and swim. Each week Shelly rides her bicycle 3 miles farther than the week before. Describe the rule that relates the number of weeks Shelly has been training to the numbers of miles she rides. Show your work, and write all fractions in simplest form.

Answer 1

Explanation:

The rule that relates the number if weeks Shelly has been training to the number of miles she rides is determined by multiplying the training weeks by 3.

Training weeks 1 2 3 4 5 6

Riding (miles) 3

If for the first training weeks; she rode her bicycle for 3 miles

Then for the second training weeks ; her riding miles will be 2 × 3 = 6 miles ( going by the rule described above )

Third week = 3 × 3 = 9 miles

Fourth week = 4 × 3 = 12 miles

Fifth week = 5 × 3 = 15 miles

Sixth week = 6 × 3 = 18 miles

Training weeks 1 2 3 4 5 6

Riding (miles) 3 6 9 12 15 18

The fractions are:

`(1)/(3) ; \ \ (2)/(6) ; \ \ (3)/(9) ; \ \ (4)/(12) ; \ \ (5)/(15) ; \ \ (6)/(18)`

To simplest form; we have:

`(1)/(3) ; \ \ (1)/(3) ; \ \ (1)/(3) ; \ \ (1)/(3) ; \ \ (1)/(3) ; \ \ (1)/(3)`

Answer 2

`a_n = 3n`

Explanation:

Each week Shelly rides her bicycle 3 miles farther than the week before.

In the 1st week she rides 3 miles.

In the 2nd week she rides 3 + 3 = 6 miles

In the 3rd week she rides 6 + 3 = 9 miles

In the 4th week she rides 9 + 3 = 12 miles

In the 5th week she rides 12 + 3 = 15 miles

So we have sequence of 3, 6, 9, 12, 15,....

The general form to represent an arithmetic sequence is given by

`a_n = a_1 + (n - 1)\cdot d`

Where
`a_n` is the nth term,
`a_1` is the first term and
`d` is the common difference.

The common difference is the difference between any two consecutive terms.

In this case we have

`a_1 = 3\\\\d = 6-3 = 3`

So the above equation becomes

`a_n = a_1 + (n - 1)\cdot d\\\\a_n = 3 + (n - 1)\cdot 3\\\\a_n = 3 + 3n - 3\\\\a_n = 3n`

So this the the required rule that relates the number of weeks (n) Shelly has been training to the numbers of miles she rides.

For example:

To find out how many miles Shelly will ride in the 10th week,

`a_n = 3n\\\\a_(10) = 3(10)\\\\a_(10) = 30\: miles\\\\`

To write in terms of fractions in simplest form

`(week)/(miles) = (10)/(30) = (1)/(3)`