Question

This is the entire question. I would like some help with all three parts, but I'm especially confused as to what they want for Part 3.
Thanks

Part 1: Create a scenario for an arithmetic sequence. For example, Jasmine practices the piano for ______ minutes on Monday. Every day she ___________ her practice time by _________. If she continues this pattern, how many minutes will she practice on the 7th day? Be sure to fill in the blanks with the words that will create an arithmetic sequence. Use your scenario to write the function for the 7th term in your sequence using sequence notation.

Part 2: Create a scenario for a geometric sequence. For example, Anthony goes to the gym for ______ minutes on Monday. Every day he _________his gym time by ____________. If he continues this pattern, how many minutes will he spend at the gym on the 5th day? Be sure to fill in the blanks with the words that will create a geometric sequence. Use your scenario to write the formula for the 5th term in your sequence, using sequence notation.

Part 3: Use your scenario from part 2 to write a question that will lead to using the geometric series formula. Use the formula to solve for Sn in your scenario.

Answer 1

1. Jasmine practices the piano for 2 minutes on Monday. Every day she increases her practice time by 2 minutes.  The formula to find a specific term is the explicit rule: $a_{n} = a_{1} + (n-1)d$ where n = the term you are looking for,  $a_{1}$ = your first term, and d = the difference between the terms.

So in this example, n=7, $a_{1} = 2$, and d=2.  Plug that into formula and your function is $a_{7} = 2 + (7-1)2 =2+6*2=2+12=14$  So on the 7th day she would practice 14 minutes, even though it didn't ask you to solve it.

2. Anthony goes to the gym for 10 minutes on Monday. Every day he increases his gym time by 10%. If he continues this pattern, how many minutes will he spend at the gym on the 5th day?

The explicit rule for geometric equations is $a_{n} = a_{1} * r^{n-1}$.
In this example, n = 5, r= 1.1, and $a_{1} = 10$. So you plug that in and you get: $a_{5} = 10* 1.1^{5-1} = 10*1.1 ^{4} =10*1.4641=14.641$

3. Anthony works out for a month with that pattern and wants to know his total minutes worked.  In this case, the formula you want is $S_{n} = \frac{ a_{1}(1- r^{n} ) }{1-r}$.  Plug in your values where the only difference is n=30.
$S_{30} = \frac{10(1-1.1 ^{5}) }{1-1.1} =1644.94$

Hope that helps