College

Select the correct type of sequence and recursive function for the sequence below:

[tex]$34, 40, 46, 52$[/tex]

A. Geometric sequence; [tex]$f(1) = 34; \, f(n) = -6 \cdot f(n-1)$[/tex], for [tex]$n \geq 2$[/tex]

B. Arithmetic sequence; [tex]$f(1) = 34; \, f(n) = f(n-1) - 6$[/tex], for [tex]$n \geq 2$[/tex]

C. Geometric sequence; [tex]$f(1) = 34; \, f(n) = 6 \cdot f(n-1)$[/tex], for [tex]$n \geq 2$[/tex]

D. Arithmetic sequence; [tex]$f(1) = 34; \, f(n) = f(n-1) + 6$[/tex], for [tex]$n \geq 2$[/tex]

Answer :

To solve the problem, first observe the given sequence:

[tex]$$34, \; 40, \; 46, \; 52$$[/tex]

1. Calculate the differences between consecutive terms:
- From [tex]$34$[/tex] to [tex]$40$[/tex]:
[tex]$$40 - 34 = 6$$[/tex]
- From [tex]$40$[/tex] to [tex]$46$[/tex]:
[tex]$$46 - 40 = 6$$[/tex]
- From [tex]$46$[/tex] to [tex]$52$[/tex]:
[tex]$$52 - 46 = 6$$[/tex]

2. Since the difference between each consecutive term is constant ([tex]$6$[/tex]), the sequence is an arithmetic sequence.

3. An arithmetic sequence has a recursive rule of the form:

[tex]$$f(1)=\text{first term} \quad \text{and} \quad f(n)=f(n-1)+d, \quad \text{for } n\geq2,$$[/tex]

where [tex]$d$[/tex] is the common difference.

4. In this case, the first term is [tex]$34$[/tex] and the common difference is [tex]$6$[/tex]. Therefore, the recursive function can be written as:

[tex]$$f(1)=34 \quad \text{and} \quad f(n)=f(n-1)+6 \quad \text{for } n\geq2.$$[/tex]

The correct answer is:

Arithmetic sequence; [tex]$$f(1)=34 \quad ; \quad f(n)=f(n-1)+6, \quad \text{for } n\geq2.$$[/tex]

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