Common ratio
Understand and calculate the common ratio of geometric sequences in Grade 9 algebra. Divide any term by the previous term to find r and use it to write explicit formulas.
Key Concepts
Property The ratio between consecutive terms is known as the common ratio . In a geometric sequence, the ratio of any term divided by the previous term is the same for any two consecutive terms.
Explanation The common ratio is the secret code of a geometric sequence! It's the one number you're always multiplying by to continue the pattern. To crack the code, just pick any two neighbors in the sequence and divide the second one by the first one. The result is your constant multiplier, whether it's a whole number, a fraction, or negative.
Examples In the sequence $4, 12, 36, 108, ...$, the common ratio is $\frac{12}{4} = 3$. For $320, 80, 20, 5, ...$, the common ratio is $\frac{ 80}{320} = \frac{1}{4}$. The signs flip because the ratio is negative! In $0.4, 1, 2.5, 6.25, ...$, the common ratio is $\frac{1}{0.4} = 2.5$.
Common Questions
What is the common ratio in a geometric sequence?
The common ratio r is the constant multiplier between consecutive terms. Divide any term by the previous: r = aₙ/aₙ₋₁. For 3, 6, 12, 24 → r = 6/3 = 2.
How do you find the nth term of a geometric sequence?
Use aₙ = a₁ × r^(n-1) where a₁ is the first term, r is the common ratio, and n is the term number.
How can you tell if a sequence is geometric?
Divide each term by the previous one. If all ratios are equal, the sequence is geometric with that common ratio.