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: 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.
write the explanations using simpler terms
Sequence: 3, 7, 15, 31, 63
Explicit formula: To find the explicit formula, we notice that each term in the sequence is obtained by doubling the previous term and adding 1. So, we can write the formula as an = 2^n + 1, where n represents the position of the term in the sequence.
Recursive formula: The recursive formula tells us how to find a term based on the previous term. In this sequence, each term is twice the previous term plus 1. So, the recursive formula is an = 2an-1 + 1, where an-1 represents the term before the current term.
By observing these patterns, we can understand how to find any term in the sequence using both the explicit and recursive formulas.