Question

Bill is able to save $35/week after working part-time and paying his expenses. These two formulas show his weekly savings: f(1) = 35, f(n) = f(1) + f(n-1) for n > 1 f(n) = 35n Which one of these formulas show the sequence written recursively and which shows it written explicitly? Justify your explanations. Use the recursive formula to make a table of values for 1 ≤ n ≤ 5. Show your calculations. Explain what your table means. Use any formula of your choice to find f(40). Explain why you chose that method and what your answer means. Show your calculations. Given the sequence of numbers: 5, 6, 8

Answer 1

f(n) = f(1) + f(n-1) for n > 1 f(n) = 35n
That's the equation that is written as a recursion formula. The reason why, is that his money grows by 35 dollars a week. There is a number called a common difference, d, that determines how much an arithmetic recursion equation is incremented by. An arithmetic recursion equation follows the form \[a _{n}=a _{1}+d(n-1)\] and looks very much like the equation they've asked you to determine. a1 is your value for the first computation of the equation. In your case, 35 dollars. d is the common difference, and n is the number of computations you are on, consider it to be weeks in this case.

For n=1-5


Really, all the table shows is how much Bill's money grows per week if he were to not spend it, in other words, his earnings.

To find f(40) we can use either the original formula f(n) = f(1) + f(n-1) for n > 1 f(n) = 35n or we can choose to use the standard arithmetic formula, which I prefer.

\[a _{40}=a _{1}=d(n-1) ; n>1\]
\[a _{40}=35+35(40-1)\]
\[a _{40}=35+35(39)\]
\[a _{40}=35+1365)\]
\[a _{40}=1400\]
This means that after 40 weeks, Bill has earned overall a total of 1400 dollars.
I'm not sure what the last part of your question is asking but hopefully you can see how recursion formulas (arithmetic sequences) are solved from my explanation.