Question

u3:04: unit 3 test
select the recursive formula for the arithmetic sequence.
26, 32, 38, 44, ...
○ next = now + 6, starting at 26
○ next = now + 6, starting at 44
○ a(n) = 44 + (n-1)(6)
○ a(n) = 26 + (n-1)(6)

Explanation:

Step1: Analyze the arithmetic sequence

The given sequence is 26, 32, 38, 44,.... The common difference \(d\) is \(32 - 26=6\), \(38 - 32 = 6\), etc. The first term \(a_1 = 26\).

Step2: Analyze recursive formula

A recursive formula for an arithmetic sequence is of the form \(a_{n}=a_{n - 1}+d\) (or in the given notation \(NEXT = NOW + d\)) with the starting value as the first term \(a_1\).

• For option 1: \(NEXT=NOW + 6\), starting at 26. The first term is 26, and each next term is previous term + 6. This matches the recursive form of arithmetic sequence.
• For option 2: Starting at 44 is wrong as the first term is 26.
• For option 3: \(A(n)=44+(n - 1)(6)\) is an explicit formula with first term 44 (since when \(n = 1\), \(A(1)=44\)) which is wrong as first term is 26.
• For option 4: \(A(n)=26+(n - 1)(6)\) is an explicit formula (not recursive) as it gives the \(n\)-th term directly in terms of \(n\), not in terms of previous term.

Answer:

A. \( \text{NEXT} = \text{NOW} + 6 \), starting at 26