Determine a recursive formula for the sequence: 2, 5, 10, 17,......
an = n ^ 2 + 1
n = 1
a1 = 1 ^ 2 + 1 = 1 + 1 = 2
n = 2
a2 = 2 ^ 2 + 1 = 4 + 1 = 5
n = 3
a3 = 3 ^ 2 + 1 = 9 + 1 = 10
n = 4
a4 = 4 ^ 2 + 1 = 16 + 1 = 17
To determine a recursive formula for the given sequence, let's analyze the pattern:
Sequence: 2, 5, 10, 17, ...
Looking closely, we can observe that each term in the sequence is obtained by adding a certain difference to the previous term.
The difference between consecutive terms follows a pattern:
5 - 2 = 3
10 - 5 = 5
17 - 10 = 7
...
The differences between consecutive terms form an arithmetic sequence with a common difference of 2.
To generate the recursive formula for the sequence, we need to express each term in terms of its preceding terms.
Let's denote the nth term in the sequence as 'a(n)'.
To determine the recursive formula, we can express 'a(n)' in terms of 'a(n-1)', the previous term in the sequence.
Based on our analysis, the recursive formula can be derived as follows:
a(n) = a(n-1) + d
In this case, 'd' represents the difference between consecutive terms, which is 2.
Therefore, the recursive formula for the given sequence is:
a(n) = a(n-1) + 2
We can use this formula to find any term in the sequence by calculating each term based on the previous term and adding 2.