Write the first five terms of a sequence. Don’t make your sequence too simple. Write both an explicit formula and a recursive formula for a general term in the sequence. Explain in detail how you found both formulas.
Sequence: 2, 5, 9, 14, 20
Explicit formula: a_n = n^2 + 1
Recursive formula:
a_1 = 2
a_n = a_(n-1) + n
Explanation:
To find the explicit formula for the sequence, we first write out the terms of the sequence and observe any patterns or relationships. In this case, we notice that each term is one more than the square of its position in the sequence. For example, the fourth term, 14, is 4^2 + 1 = 16 + 1 = 17. Therefore, the explicit formula is a_n = n^2 + 1.
To find the recursive formula for the sequence, we consider how each term in the sequence is related to the previous term. In this case, each term is equal to the previous term plus the position of the term in the sequence. For example, the fifth term, 20, is obtained by adding 5 to the previous term, 14. Therefore, the recursive formula is a_n = a_(n-1) + n.