Question

identify each sequence as arithmetic or geometric.
move each sequence to the correct category.
arithmetic sequence geometric sequence
4, 12, 36, 108, ... 100, 90, 80, 70, ... 120, 60, 30, 15, ... 3, 8, 13, 18, ...

Explanation:

Step1: Recall Definitions

An arithmetic sequence has a common difference (\(d\)) between consecutive terms, where \(d = a_{n+1}-a_n\). A geometric sequence has a common ratio (\(r\)) between consecutive terms, where \(r=\frac{a_{n+1}}{a_n}\).

Step2: Analyze \(4, 12, 36, 108, \dots\)

Calculate the ratio between terms: \(\frac{12}{4} = 3\), \(\frac{36}{12}=3\), \(\frac{108}{36}=3\). Common ratio \(r = 3\), so it's geometric.

Step3: Analyze \(100, 90, 80, 70, \dots\)

Calculate the difference: \(90 - 100=- 10\), \(80 - 90=-10\), \(70 - 80=-10\). Common difference \(d=-10\), so it's arithmetic.

Step4: Analyze \(120, 60, 30, 15, \dots\)

Calculate the ratio: \(\frac{60}{120}=\frac{1}{2}\), \(\frac{30}{60}=\frac{1}{2}\), \(\frac{15}{30}=\frac{1}{2}\). Common ratio \(r=\frac{1}{2}\), so it's geometric.

Step5: Analyze \(3, 8, 13, 18, \dots\)

Calculate the difference: \(8 - 3 = 5\), \(13 - 8 = 5\), \(18 - 13 = 5\). Common difference \(d = 5\), so it's arithmetic.

Answer:

• Arithmetic Sequence: \(100, 90, 80, 70, \dots\); \(3, 8, 13, 18, \dots\)
• Geometric Sequence: \(4, 12, 36, 108, \dots\); \(120, 60, 30, 15, \dots\)