What Are Geometric Sequences?
Geometric sequences are special number patterns where each term is found by
multiplying the previous term by the same number (called the common ratio).
When we use fractions as the common ratio, the sequence gets smaller each step - like magic shrinking
numbers! These patterns help us understand how things grow or shrink by the same fraction each time.
How to Find the Next Term
1️⃣ Identify the common ratio (what we multiply by each time)
2️⃣ Multiply the last term by the common ratio
3️⃣ Simplify the fraction if needed
Let's Try Some Examples!
Example 1: The Shrinking Sequence
Find the next 3 terms in this sequence: 64, 16, 4, 1, ...
The common ratio is \(\frac{1}{4}\) because:
\(16 \div 64 = \frac{1}{4}\),
\(4 \div 16 = \frac{1}{4}\),
\(1 \div 4 = \frac{1}{4}\)
The next 3 terms are:
\(1 \times \frac{1}{4} = {\frac{1}{4}}\)
\(\frac{1}{4} \times \frac{1}{4} = {\frac{1}{16}}\)
\(\frac{1}{16} \times \frac{1}{4} = {\frac{1}{64}}\)
Example 2: The Pizza Sequence
Imagine you have 1 whole pizza. Each step, you eat half of what remains. How much pizza is left after 4 steps?
Start: 1 whole pizza
After 1st bite: \(\frac{1}{2}\) remains (you ate \(\frac{1}{2}\))
After 2nd bite: \(\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}\) remains
After 3rd bite: \(\frac{1}{4} \times \frac{1}{2} = \frac{1}{8}\) remains
After 4th bite: \(\frac{1}{8} \times \frac{1}{2} = \frac{1}{16}\) remains
Only \(\frac{1}{16}\) of the pizza is left after 4 bites!
Parent Tips 🌟
- Kitchen Math: Use measuring cups to demonstrate geometric sequences (½ cup, ¼ cup, ⅛ cup) to show real-world applications.
- Visual Aids: Draw rectangles and keep shading half of the remaining white space to visualize the shrinking sequence.
- Game Time: Play "Sequence Detective" where you start a geometric sequence with fractions and have your child find the pattern.