Question

in an arithmetic sequence with a common difference of -4, which of the following is true?
a. the terms decrease by 4 each time.
b. the terms increase by 4 each time.
c. the terms stay constant.
d. the terms alternate between positive and negative.

Explanation:

Step1: Recall arithmetic sequence definition

In an arithmetic sequence, the common difference \( d \) is the difference between consecutive terms (\( a_{n + 1}-a_n=d \)).

Step2: Analyze the common difference of -4

If \( d=-4 \), then \( a_{n + 1}=a_n + (-4)=a_n - 4 \). This means each term is 4 less than the previous term, so the terms decrease by 4 each time.

Step3: Evaluate other options

• Option b: A positive common difference would mean increase, but \( d=-4 \) is negative, so b is false.
• Option c: Terms stay constant only if \( d = 0 \), not -4, so c is false.
• Option d: The sign of terms depends on the first term and common difference, but a common difference of -4 doesn't guarantee alternating signs (e.g., starting at 1: 1, -3, -7,... all negative after first term), so d is false.

Answer:

a. The terms decrease by 4 each time.