Question

when an arithmetic sequence decreases, the common difference d is .

Explanation:

Step1: Recall Arithmetic Sequence Definition

An arithmetic sequence has the form \( a_n = a_1 + (n - 1)d \), where \( a_1 \) is the first term, \( n \) is the term number, and \( d \) is the common difference.

Step2: Analyze Decreasing Sequence

For a sequence to be decreasing, each term \( a_{n+1} \) must be less than \( a_n \). So \( a_{n+1}-a_n<0 \). But \( a_{n+1}-a_n = d \) (by the arithmetic sequence formula, since \( a_{n+1}=a_1 + nd \) and \( a_n=a_1+(n - 1)d \), subtracting gives \( d \)). Thus, if the sequence is decreasing, \( d<0 \), meaning the common difference is negative.

Answer:

negative