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: [tex] a_{n} = a_{1} + (n-1)d[/tex] where n = the term you are looking for, [tex] a_{1} [/tex] = your first term, and d = the difference between the terms.
So in this example, n=7, [tex] a_{1} = 2[/tex], and d=2. Plug that into formula and your function is [tex] a_{7} = 2 + (7-1)2 =2+6*2=2+12=14[/tex] 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 [tex] a_{n} = a_{1} * r^{n-1} [/tex].
In this example, n = 5, r= 1.1, and [tex] a_{1} = 10[/tex]. So you plug that in and you get: [tex] a_{5} = 10* 1.1^{5-1} = 10*1.1 ^{4} =10*1.4641=14.641[/tex]
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 [tex] S_{n} = \frac{ a_{1}(1- r^{n} ) }{1-r} [/tex]. Plug in your values where the only difference is n=30.
[tex] S_{30} = \frac{10(1-1.1 ^{5}) }{1-1.1} =1644.94[/tex]
Hope that helps
