Write a recursive formula for the sequence: 16, 10, 7, 5.5, 4.75...
Given that \(a_1 = 16\), what is the recursive formula for this sequence?
Answer (3)
a 1 = 2 1 24 + 4 = 16 a 2 = 2 2 24 + 4 = 10 a 3 = 2 3 24 + 4 = 7 a 4 = 2 4 24 + 4 = 5.5 a 5 = 2 5 24 + 4 = 4.75 a 6 = 2 6 24 + 4 = 4.375 ⋮ a n = 2 n 24 + 4
The recursive formula for the sequence is:
an = 24/(2ⁿ)+ 4, for n > 1
What is a sequence?
An ordered list of things is a sequence (or events). Similar to a set, it has members (also called elements , or terms). The length of the sequence is the number of ordered items (potentially infinite).
To find the recursive formula for the sequence 16, 10, 7, 5.5, 4.75..., we need to find the rule that generates each term based on the previous terms.
That is:
a1 = 24/2¹ + 4 = 12 + 4 = 16
a2 = 24/2² + 4 = 6 + 4 = 10
a3 = 24/2³ +4 = 3 + 4 =7
a4 = 24/2⁴ + 4 = 5.5
a5 = 24/2⁵ + 4 = 4.75
We can see that the formula for the sequence is:
a1 = 16
an = 24/(2ⁿ)+ 4, for n > 1
Therefore, the recursive formula for the sequence is:
an = 24/(2ⁿ)+ 4, for n > 1
where a1 is the first term, and each subsequent term is obtained by dividing 24 with the previous term by 2ⁿ and then adding 4.
To learn more about the sequence ;
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The recursive formula for the sequence 16, 10, 7, 5.5, 4.75... is defined as a 1 = 16 and a n = a n − 1 − ( 6 × ( 0.5 ) n − 2 ) for 1"> n > 1 . This formula shows how each term is derived by subtracting a diminishing value from the previous term. The initial term is given by a 1 .
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