Lesson 105: Recognizing and Extending Geometric Sequences

New Concept

A geometric sequence is a sequence with a constant ratio between consecutive terms.

What’s next

Next, you’ll practice finding the common ratio and use it to extend sequences and calculate any term with a powerful formula.

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 $512 \times 4 = 2048$.

Let's see how a single multiplier generates an entire sequence, even with alternating signs. This example shows how to extend a sequence, a key idea from this lesson.

Example Problem

Find the next four terms in the geometric sequence $3, -12, 48, -192, ...$

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 $\frac{1}{0.4} = 2.5$.

Property

Let $A(n)$ equal the $n$th term of a geometric sequence, then

Explanation

Want to jump to any term in a sequence without listing them all? This formula is your mathematical time machine! Just plug in the first term ($a$), the common ratio ($r$), and the term number you want ($n$). Remember the tricky part: the exponent is always one less than the term number ($n-1$) because the first term doesn't get multiplied yet!

Examples

• Find the 5th term if $a=6$ and $r=-2$: $A(5) = 6(-2)^{5-1} = 6(-2)^4 = 6(16) = 96$.
• Find the 8th term of $\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, ...$. Here $a=\frac{1}{2}$ and $r=\frac{1}{2}$. $A(8) = \frac{1}{2}(\frac{1}{2})^{8-1} = (\frac{1}{2})^8 = \frac{1}{256}$.
• A ball's first bounce is 1.5 yards high ($a=1.5$), and each bounce is 80% of the last ($r=0.8$). The 5th bounce height is $A(5) = 1.5(0.8)^{5-1} \approx 0.61$ yards.