Question

You bike 1 mile the first day of your​ training, 1.3 miles the second​ day, 1.9 miles the third​ day, and 3.1 miles the fourth day. If you continue this​ pattern, how many miles do you bike the seventh​ day?

On the seventh day you bike ____miles.

Answer 1

19.9 miles

Explanation:

In this problem we have:

`d_1=1 mi` is the distance travelled during the 1st day

`d_2=1.3 mi` is the distance travelled during the 2nd day

`d_3=1.9 mi` is the distance travelled during the 3rd day

`d_4=3.1 mi` is the distance travelled during the 4th day

We notice that the difference between the distance travelled on the (n+1)-th day and the distance travelled on the n-th day doubles every day. In fact:

`d_2-d_1=0.3\\d_3-d_2=2\cdot 0.3 = 0.6\\d_4-d_3=2\cdot 0.6 = 1.2`

Which can be rewritten using the general formula:

`d_(n+1)-d_n=2(d_n-d_(n-1))`

This means that

`d_(n+1)=d_n+2(d_n-d_(n-1))`

By applying this formula recursively, we can find the 7th term, which is the distance travelled on the 7th day:

`d_1=1\\d_2=1.3\\d_3=1.9\\d_4=3.1\\d_5=3.1+2\cdot 1.2=5.5\\d_6=5.5+2\cdot 2.4=10.3\\d_7=10.3+2\cdot 4.8=19.9 mi`

So, the distance travelled on the 7th day is 19.9 miles.