Renee belongs to a bowling club. She scores 50, 52, 55, and 59 on her first four games. She hopes to continue improving according to this pattern.

Part A: What are the next four terms in Renee’s sequence?
Part B: Write an recursive function rule for the sequence?
Part C: What is the domain and range of your function rule?
Part D: Do you think Renee can keep up this pattern? Explain.

terms differ by 2,3,4,...

2+3+4+...+n = n(n+1)/2-1
so, the sequence is

Tn = 49 + n(n+1)/2
50,52,55,59,64,70,77,85,...

T1 = 50
Tn = T(n-1) + n

T21 = 280
T22 = 302

So, she can't keep it up past 21 games.

To find the next four terms in Renee's sequence, we can observe the pattern in her scores. The difference between each consecutive score is increasing by 2.

Part A: The sequence starts with 50, 52, 55, 59. To find the next four terms, we can continue adding 2 to each consecutive term:
61 (59 + 2 = 61), 63 (61 + 2 = 63), 65 (63 + 2 = 65), 67 (65 + 2 = 67).

Therefore, the next four terms in Renee's sequence are 61, 63, 65, and 67.

Part B: To write a recursive function rule for the sequence, we can use the fact that each term is obtained by adding 2 to the previous term. Let's call the function f(n), where n represents the position of the term in the sequence (starting from 1). The rule can be written as:
f(n) = f(n-1) + 2, where n > 1 and f(1) = 50.

For example, f(2) = f(1) + 2 = 50 + 2 = 52.
f(3) = f(2) + 2 = 52 + 2 = 54.

Part C: The domain of the function rule is the set of positive integers since n represents the position in the sequence. The range is the set of scores in the sequence, which are positive integers starting from 50.

Domain: {1, 2, 3, ...}
Range: {50, 52, 54, 56, ...}

Part D: Based on the given pattern, it seems that Renee is consistently improving her scores by 2 each time. However, it's important to note that maintaining the pattern indefinitely is not guaranteed. Factors such as skill level, practice, form, and external conditions can impact a bowler's performance. Therefore, it is possible for Renee to deviate from the pattern in the future.