To determine the recursive formula for the sequence for the term numbers 1, 2, 3, 4, and 5, we need to analyze the options given and see which one produces a valid sequence where each term is dependent on previous terms.
The provided options appear to represent different forms of recursive relationships:
- \( a_n = 1; , a_n = a_{n-1} + a_{n-1} \)
- \( a_n = 1; , a_n = a_{n-1} + 1 \)
- \( a_n = 1; , a_n = a_{n-1} + 2 \)
- \( a_n = 1; , a_n = a_{n-1} + a_{n+1} \)
We need to consider various approaches for each option:
- The first option implies duplication: if \( a_n = a_{n-1} + a_{n-1} \), this can lead to terms growing very rapidly.
- The second option, \( a_n = a_{n-1} + 1 \), gives a linear growth which is simple and straightforward.
- The third option, \( a_n = a_{n-1} + 2 \), offers an even faster growth than option two.
- The fourth option, \( a_n = a_{n-1} + a_{n+1} \), is not a valid recursive formula since it relies on future terms.
To decide which of the equations ultimately produce common sequences, let's look deeper into the implications of each form step by step.
Assuming \( a_1 = 1 \):
- For the second option:
- \( a_1 = 1 \)
- \( a_2 = a_1 + 1 = 1 + 1 = 2 \)
- \( a_3 = a_2 + 1 = 2 + 1 = 3 \)
- \( a_4 = a_3 + 1 = 3 + 1 = 4 \)
- \( a_5 = a_4 + 1 = 4 + 1 = 5 \)
For the second option, you arrive at a sequence of 1, 2, 3, 4, 5.
Thus, depending on the options, the recursive formula that accurately represents this growth is:
an=1; an=an−1+1
This provides a simple linear growth.