Question

A hiker on the Appalachian Trail planned to increase the distance covered by 10% each day. After 7 days, the total distance traveled is 75.897 miles.What is the equation for Sn? Show all necessary math work.

Answer 1

We know that the distance he covers increases by 10% each day.

Then, being an the distance covered in day n, we have the recursive equation:

$a_n=(1+\frac{10}{100})\cdot a_{n-1}=1.1\cdot a_{n-1}$

This is a geometric series with r = 1.1.

It will have the explicit formula:

$a_n=a_1\cdot r^{n-1}$

If after 7 days, the total distance travelled is 75897 miles, we can use the formula for the sum of terms in geometric series like this:

$S_n=\sum ^n_{k\mathop=0}a_k=\sum ^n_{k\mathop{=}0}a_1r^{k-1}=a_1\frac{1-r^n}{1-r}$

As we know that when n = 7, Sn = 75897 and r is r = 1.1, we can find a1 as:

$\begin{gathered} S_7=75897 \\ a_1\cdot\frac{1-1.1^7}{1-1.1}=75897 \\ a_1\cdot\frac{1-1.9487171}{-0.1}=75897 \\ a_1\cdot\frac{-0.948771}{-0.1}=75897 \\ a_1\cdot9.487171=75897 \\ a_1=\frac{75897}{9.487171} \\ a_1\approx8000 \end{gathered}$

Then, we can now use a1 to find a complete expression for Sn:

$S_n=8000\cdot\frac{1-1.1^n}{1-1.1}=-\frac{8000}{0.1}\cdot(1-1.1^n)=-80000\cdot(1-1.1^n)$

We can test this for n = 7:

$\begin{gathered} S_7=-80000\cdot(1-1.1^7) \\ S_7=-80000\cdot(1-1.9487171) \\ S_7=-80000\cdot(-0.9487171) \\ S_7\approx75897 \end{gathered}$

Answer: the expression for the sum of n terms, Sn, is Sn = -80000*(1 - 1.1^n)