Common Ratio Worksheet/Game

Common Ratio Worksheet/Game
Welcome to Common Ratio Worksheet/Game. A geometric sequence is a list of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio (r). This game requires you find the common ratio and the next term in the geometric sequence. Scroll down the page for a more detailed explanation.

 

 

How to play the Common Ratio Game
Phase 1: Identify the Ratio (r) Your first task is to figure out what number is being multiplied to get from one term to the next.

1. Analyze the Direction:
   If numbers are getting larger (e.g., 4, 12, 36), r is a whole number (in this case, 3).
   If numbers are getting smaller (e.g., 80, 40, 20), r is a fraction (1/2) or decimal (0.5).
   
2. The Division Trick:
   Divide the second number by the first number (n₂ ÷ n₁).
   Example: If n₁ = 100 and n₂ = 20, then 20 / 100 = 0.2 (or 1/5).
   Flexible Entry: You can type your answer as a fraction like 1/4 or as a decimal like 0.25.
   

How to find the Ratio (r)
If you are looking at a sequence and aren’t sure what the ratio is, pick any term and divide it by the term before it:
\(r = \frac{a_n}{a_{n-1}}\)
Example: In the sequence 5, 20, 80, 320, …
Calculate: 20 ÷ 5 = 4.
Check: 80 ÷ 20 = 4.
The common ratio is 4.

Phase 2: Predict n₅
Once the ratio is verified, you must calculate the very next number in the pattern.

1. Look at n₄: This is the last number visible on the cards.
   
2. Apply the Ratio: Multiply n₄ by the r value you just found.
   Example (Growth): n₄ = 54, r = 3. Calculation: 54 × 3 = 162.
   Example (Shrinking): n₄ = 4, r = 1/2. Calculation: 4 × 0.5 = 2.
   
3. Decimal Precision: If your calculation results in a decimal (like 0.8), enter it exactly as it appears.
   

What is a Geometric Sequence?
A geometric sequence is a list of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio (r).

1. The Core Components
   To define any geometric sequence, you only need two pieces of information:
   The First Term (a₁): Where the sequence starts.
   The Common Ratio (r): The multiplier used to move from one term to the next.
   

2. Growth vs. Decay
   The behavior of the sequence depends entirely on the value of the ratio r:
   

   If the Ratio is…	The Sequence…Example
   r > 1	Grows (Exponential Growth)	2, 6, 18, 54, … (r=3)
   0 < r < 1	Shrinks (Exponential Decay)	100, 50, 25, 12.5, … (r=0.5)
   r < 0	Oscillates (Switches signs)	4, -8, 16, -32,… (r=-2)

Common Ratio of an Geometric Sequence

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