Number Sequence Calculator · Practical guide
Geometric Sequence with a Negative Common Ratio
A negative ratio makes terms flip sign every step — the formulas still work exactly the same.
At a glance
What a Negative Ratio Does
When the common ratio r is negative, a geometric sequence alternates between positive and negative terms. With a₁ = 5 and r = −2, the sequence runs 5, −10, 20, −40, 80, −160, … — each term flips sign because multiplying by a negative number reverses it.
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Number Sequence Calculator
Enter your own values, then compare the result with the examples below.
Arithmetic Sequence Result
First 10 terms:
Geometric Sequence Result
First 10 terms:
Nth Term with a Negative Ratio
The standard formula aₙ = a₁ × rⁿ⁻¹ still applies. For a₁ = 5 and r = −2, the 10th term is a₁₀ = 5 × (−2)⁹ = 5 × (−512) = −2560. The even powers of r give positive terms and odd powers give negative ones — that is the whole pattern.
Sum of an Alternating Sequence
The sum formula Sₙ = a₁(1 − rⁿ)/(1 − r) also works unchanged. For the first 10 terms above: S₁₀ = 5 × (1 − (−2)¹⁰)/(1 − (−2)) = 5 × (1 − 1024)/3 = 5 × (−1023)/3 = −1705. Notice the sum can be negative even though every other term is positive, because the later negative terms outweigh the positive ones.
Special Cases Worth Knowing
- r = −1: the sequence just bounces between two values — a₁, −a₁, a₁, −a₁, … — and the sum alternates between a₁ and 0
- r between −1 and 0: terms alternate while shrinking toward zero, like 8, −4, 2, −1, 0.5, …
- r = 1: every term equals the first term, and the sum is simply n × a₁
Select the Geometric tab in the calculator above and enter a negative ratio to see the alternating term list and exact sums for your own sequence.
Frequently Asked Questions
Is a sequence with a negative ratio still called geometric?
Yes. The definition only requires each term to be the previous term multiplied by a constant ratio — that ratio can be positive, negative, or fractional.
How do I find the common ratio of an alternating sequence?
Divide any term by the one before it. For 5, −10, 20, −40: −10 ÷ 5 = −2, and 20 ÷ −10 = −2, so r = −2. The negative sign is part of the ratio.
Can an alternating geometric sequence converge?
Yes, when the ratio's absolute value is less than 1 — for example r = −0.5. The terms alternate in sign but shrink toward zero, and the infinite sum converges to a₁/(1 − r).