Question

1. classify

because the terms of the sequence (a) ____ , the sequence is (b) ____ .

(a)

• have a common difference
• have a common ratio
• are increasing
• are decreasing

(b)

• arithmetic
• geometric
• neither arithmetic nor geometric

1. write recursive formula

Explanation:

To solve this, we recall the definitions of arithmetic and geometric sequences:

• An arithmetic sequence is defined by terms with a common difference (the difference between consecutive terms is constant).
• A geometric sequence is defined by terms with a common ratio (the ratio between consecutive terms is constant).

Step 1: Analyze Option (A)

For (A), we choose the characteristic that defines the sequence type:

• If the sequence is arithmetic, (A) is "have a common difference".
• If the sequence is geometric, (A) is "have a common ratio".
• "are increasing" or "are decreasing" do not define the type of sequence (arithmetic/geometric), only its trend.

Step 2: Analyze Option (B)

For (B), we match the (A) characteristic to the sequence type:

• If (A) is "have a common difference", then (B) is "arithmetic".
• If (A) is "have a common ratio", then (B) is "geometric".
• If neither, (B) is "neither arithmetic nor geometric".

Example (Assuming an Arithmetic Sequence)

Suppose the sequence is arithmetic (e.g., \( 2, 5, 8, 11, \dots \), where the common difference is \( 3 \)):

• (A) would be "have a common difference".
• (B) would be "arithmetic".

Example (Assuming a Geometric Sequence)

Suppose the sequence is geometric (e.g., \( 2, 6, 18, 54, \dots \), where the common ratio is \( 3 \)):

• (A) would be "have a common ratio".
• (B) would be "geometric".

Final Answer (for a typical arithmetic sequence example)

(A) \(\boldsymbol{\text{have a common difference}}\)
(B) \(\boldsymbol{\text{arithmetic}}\)

(Note: If the sequence is geometric, (A) is "have a common ratio" and (B) is "geometric". If neither, (A) would not be "have a common difference" or "have a common ratio", and (B) is "neither arithmetic nor geometric".)

Answer:

To solve this, we recall the definitions of arithmetic and geometric sequences:

• An arithmetic sequence is defined by terms with a common difference (the difference between consecutive terms is constant).
• A geometric sequence is defined by terms with a common ratio (the ratio between consecutive terms is constant).

Step 1: Analyze Option (A)

For (A), we choose the characteristic that defines the sequence type:

• If the sequence is arithmetic, (A) is "have a common difference".
• If the sequence is geometric, (A) is "have a common ratio".
• "are increasing" or "are decreasing" do not define the type of sequence (arithmetic/geometric), only its trend.

Step 2: Analyze Option (B)

For (B), we match the (A) characteristic to the sequence type:

• If (A) is "have a common difference", then (B) is "arithmetic".
• If (A) is "have a common ratio", then (B) is "geometric".
• If neither, (B) is "neither arithmetic nor geometric".

Example (Assuming an Arithmetic Sequence)

Suppose the sequence is arithmetic (e.g., \( 2, 5, 8, 11, \dots \), where the common difference is \( 3 \)):

• (A) would be "have a common difference".
• (B) would be "arithmetic".

Example (Assuming a Geometric Sequence)

Suppose the sequence is geometric (e.g., \( 2, 6, 18, 54, \dots \), where the common ratio is \( 3 \)):

• (A) would be "have a common ratio".
• (B) would be "geometric".

Final Answer (for a typical arithmetic sequence example)

(A) \(\boldsymbol{\text{have a common difference}}\)
(B) \(\boldsymbol{\text{arithmetic}}\)

(Note: If the sequence is geometric, (A) is "have a common ratio" and (B) is "geometric". If neither, (A) would not be "have a common difference" or "have a common ratio", and (B) is "neither arithmetic nor geometric".)