Question

In preparing for a marathon run in the spring, Joelle runs 5 miles in week four of training and 10 miles in week 14. Joelle’s weekly increase in mileage follows an arithmetic sequence.
a. Write the general term that relates the number of miles to the week number of training.


b. An official marathon is 26 miles long. In which week would Joelle run at least 26 miles?

Answer 1

Given: $a_4=5; a_{14}=10; a_k=26.$

Arithmetic sequence: $a_n=a_1+d(n-1).$

$$\begin{cases}a_4=a_1+3d=5 \\ a_{14}=a_1+13d=10 \end{cases}$$

Solving the system, we get

$$\begin{cases}a_1=3.5 \\ d=0.5 \end{cases}$$

a. General term: $a_n=3.5+0.5(n-1)=0.5n+3$

$$a_k=0.5k+3=26 \rightarrow k=46$$

b. At forty-sixth week she will be able to run 26 miles