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+(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(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(1-1.1^7)/(1-1.1)=75897 \\ a_1\cdot(1-1.9487171)/(-0.1)=75897 \\ a_1\cdot(-0.948771)/(-0.1)=75897 \\ a_1\cdot9.487171=75897 \\ a_1=(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(1-1.1^n)/(1-1.1)=-(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)