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.
Thank you
Part A:
To find the next four terms in Renee's sequence, we need to identify the pattern in her scores. Looking at the given scores (50, 52, 55, and 59), we can observe that Renee's scores are increasing by 2, 3, and 4 in each consecutive game.
Based on this pattern, the next four terms can be calculated as follows:
- The fifth term = the fourth term + 2 = 59 + 2 = 61
- The sixth term = the fifth term + 3 = 61 + 3 = 64
- The seventh term = the sixth term + 4 = 64 + 4 = 68
- The eighth term = the seventh term + 5 = 68 + 5 = 73
So, the next four terms in Renee's sequence are 61, 64, 68, and 73.
Part B:
To write a recursive function rule for the sequence, we need to find the relationship between the terms in the sequence.
Let's denote the first term as a(1), the second term as a(2), and so on. We can observe that each term can be obtained by adding a constant value to the previous term.
The recursive function rule for the sequence can be written as:
a(n) = a(n-1) + (n-1)
Here, a(n) represents the nth term in the sequence, and a(n-1) represents the previous term. The value (n-1) represents the number of games played before the current game.
Part C:
The domain of this function rule is the set of positive integers, as it represents the terms of a sequence where the number of games played can only be a positive whole number.
The range of this function rule is the set of scores that can be obtained by following the pattern. In this case, since Renee starts with a score of 50 and each subsequent score is the sum of the previous score and the number of games played minus one, the range will be all the possible scores that can be obtained by following this pattern.
Part D:
Based on the pattern observed, Renee's scores are improving by an increasing amount in each consecutive game. However, it is difficult to say whether Renee can keep up this pattern indefinitely. It is possible that external factors or personal performance may affect her scores in future games.
Therefore, it is uncertain whether Renee can continue improving according to this pattern in the long term. The pattern may continue for a few more games, but it is not guaranteed to persist indefinitely.