What is the recursive formula for this sequence?
12, 16, 20, 24, 28,…
OA
B.
C.
D.
4
a = 3-1 +12
a₁ = 12
a= an-1 +4
3₁= 32
an = 3n-1 +4
a = 12
The Correct Answer and Explanation is :
The given sequence is: 12, 16, 20, 24, 28,…
Identifying the Recursive Formula
To find the recursive formula, we first observe the pattern in the sequence. We notice that each term is obtained by adding 4 to the previous term:
- ( a_1 = 12 ) (the first term)
- ( a_2 = a_1 + 4 = 12 + 4 = 16 )
- ( a_3 = a_2 + 4 = 16 + 4 = 20 )
- ( a_4 = a_3 + 4 = 20 + 4 = 24 )
- ( a_5 = a_4 + 4 = 24 + 4 = 28 )
From this, we can see that the recursive relationship can be defined as follows:
- Base Case: ( a_1 = 12 )
- Recursive Step: ( a_n = a_{n-1} + 4 ) for ( n > 1 )
Correct Answer
Thus, the correct recursive formula for the sequence is:
- ( a_1 = 12 )
- ( a_n = a_{n-1} + 4 ) for ( n \geq 2 )
Explanation of the Recursive Formula
A recursive formula expresses each term of a sequence based on its preceding terms. In this case, the first term is explicitly defined as ( a_1 = 12 ). The subsequent terms are derived by consistently adding 4 to the previous term, which highlights the linear nature of this sequence.
This formula captures the essence of recursion, where each term’s calculation relies solely on the prior term, allowing for easy computation of any term in the sequence without needing to derive a closed formula. For instance, if we want to calculate the sixth term, ( a_6 ), we can use the recursive formula:
- ( a_6 = a_5 + 4 = 28 + 4 = 32 )
This approach makes recursive formulas particularly useful in programming and algorithm design, where problems can often be broken down into smaller subproblems. The recursive nature of the sequence also suggests that it can be efficiently computed through iteration or recursion in computer science contexts. In summary, the recursive formula effectively captures the relationship between consecutive terms while providing a straightforward method to generate terms in the sequence.