Twelve flags are evenly spaced around a running track. Ryan started running at the first flag and took 30 seconds to reach the sixth flag. How many seconds did it take Ryan, running at a constant rate, to reach the nth flag for the mth time?

The time is seconds to reach the n-th flag.

The time needed by Ryan (), in seconds, to reach the n-th flag at constant velocity is modelled by the following arithmetic progression:

(1)

Where:

And the period can be derived from (1): (, )

If we know that , and , then we have the following formula:

The time is seconds to reach the n-th flag.

To learn more on arithmetic progressions, we kindly invite to check this verified question: link

Ryan takes 6n+36m -42 seconds to reach the nth flag for the mth time.

Step-by-step explanation:

It takes Ryan to run from 1st to 6th flag in 30 seconds, so it takes him

30 * 6/3 = 36 seconds to make one complete round.

or it takes 6 seconds to run from one flat to the next.

To reach the nth flag (n=1,2,3,4,5, or 6)

Ryan takes 6(n-1) seconds.

To reach it the mth times, he needs to add 36(m-1) seconds.

So time it takes Ryan to reach the nth flag for the mth time takes

6(n-1) + 36(m-1)

= 6n - 6 + 36m - 36

= 6n+36m -42 seconds

27

step-by-step explanation: