Sequence: 3, 7, 15, 31, 63
Explicit formula: an = 2^n + 1
In order to find the explicit formula for this sequence, we need to observe the pattern in the terms. It is evident that each term is obtained by multiplying the previous term by 2 and adding 1. Starting with the first term, which is 3, we can see the following pattern:
3 * 2 + 1 = 7
7 * 2 + 1 = 15
15 * 2 + 1 = 31
31 * 2 + 1 = 63
Based on this pattern, we can derive the explicit formula an = 2^n + 1, where n represents the position of the term in the sequence.
Recursive formula: an = 2an-1 + 1
To find the recursive formula for the general term in the sequence, we need to analyze how each term is related to the previous term. In this case, each term is obtained by multiplying the previous term by 2 and adding 1. Therefore, the recursive formula can be expressed as an = 2an-1 + 1, where an-1 represents the previous term in the sequence.
By following this process, we can derive both the explicit and recursive formulas for the given sequence.