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 rercursively and which shows it written explicity? Justify your explanantions. Use the recursive formula to make a table of values for 1 ≤ n ≤ 5. Show your calculations. Use the explicit formula to demonstrate the most direct method 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, 11, 15, 20, 26, 33, 41, ... Explain whether or not this sequence can be considered a function.

1. We use the recursive formula to make the table of values:

f (1) = 35

f (2) = f (1) + f (2-1) = f (1) + f (1) = 35 + 35 = 70

f (3) = f (1) + f (3-1) = f (1) + f (2) = 35 + 70 = 105

f (4) = f (1) + f (4-1) = f (1) + f (3) = 35 + 105 = 140

f (5) = f (1) + f (5-1) = f (1) + f (4) = 35 + 140 = 175

2. We observe that the pattern is that for each increase of n by 1, the value of f (n) increases by 35. The explicit equation would be that f (n) = 35n. This fits with the description that Bill saves up $35 each week, thus meaning that he adds $35 to the previous week's value.

3. Therefore, the value of f (40) = 35*40 = 1400. This is easier than having to calculate each value from f (1) up to f (39) individually. The answer of 1400 means that Bill will have saved up $1400 after 40 weeks.

4. For the sequence of 5, 6, 8, 11, 15, 20, 26, 33, 41 ...

The first-order differences between each pair of terms is: 1, 2, 3, 4, 5, 6, 7, 8 ... since these differences form a linear equation, this sequence can be expressed as a quadratic equation. Since quadratics are functions (they do not have repeating values of the x-coordinate), therefore, this sequence can also be considered a function.