Descending arithmetic sequences: zero is a term, not a stopping rule
Use a negative common difference and check term counts when a sequence crosses zero.
Method
An arithmetic sequence changes by a fixed amount at every step. With a negative common difference d, it decreases. The first term has index 1, so reaching term n requires n−1 changes. Crossing zero does not stop the mathematical sequence. Whether negative values make sense depends on what the numbers represent.
Worked example
Enter first term 18, common difference −3 and eight terms in Arithmetic sequence. The eighth term is −3 and the sum is 60. The explicit list below contains zero as the seventh term. Counting only the positive terms would give six terms and a different sum.
a = 18; d = −3; n = 8
a8 = 18 + (8 − 1) × (−3) = −3
18, 15, 12, 9, 6, 3, 0, −3
S8 = 8 × (18 + (−3)) / 2 = 60
n = 1 − a / d = 7
Checks and limits
The sum can be checked by multiplying the term count by the average of the first and last terms. To locate a zero term when d is nonzero, solve 1−a/d. It must be a positive integer to identify an actual term; otherwise the sequence passes between zero’s neighbouring values. Stock counts cannot become negative without a changed real-world interpretation.
Related calculators
Source: OpenStax