Geometric sequence
Identify and extend geometric sequences in Grade 9 algebra by recognizing a constant ratio between consecutive terms and applying the formula aₙ=a₁·r^(n-1) to find any term.
Key Concepts
Property A geometric sequence is a sequence with a constant ratio between consecutive terms. The ratio between consecutive terms is known as the common ratio .
Explanation Think of it as a number pattern where you multiply by the same secret number to get from one term to the next! This special multiplier, called the common ratio, is the key to the whole sequence. To find it, just divide any term by the one that came right before it. It’s a chain reaction of multiplication!
Examples In the sequence $3, 15, 75, ...$, find the common ratio ($\frac{15}{3}=5$) and multiply to find the next term: $75 \times 5 = 375$. For $100, 50, 25, ...$, the ratio is $ \frac{1}{2}$. The next term is $25 \times ( \frac{1}{2})= 12.5$. The sequence $2, 8, 32, 128, ...$ has a common ratio of 4. The next two terms are $128 \times 4 = 512$ and $512 \times 4 = 2048$.
Common Questions
What is a geometric sequence?
A geometric sequence is a list of numbers where each term is found by multiplying the previous term by a fixed constant called the common ratio r. For example, 2, 6, 18, 54 has a common ratio of 3.
How do you find the common ratio of a geometric sequence?
Divide any term by the term before it. If the result is the same throughout the sequence, that value is the common ratio r. For 5, 15, 45, 135: 15/5 = 3, 45/15 = 3, so r = 3.
What is the formula for the nth term of a geometric sequence?
The nth term is aₙ = a₁ · r^(n-1), where a₁ is the first term, r is the common ratio, and n is the term number. For the sequence starting at 4 with r = 2, the 5th term is 4 · 2⁴ = 64.