Show that 6, 36, 216 are continued proportion.

Answer

To show that the numbers 6, 36, and 216 are in continued proportion, we can follow these steps:

### Step 1: Understand the Definition of Continued Proportion
Continued proportion means that for three numbers \( a, b, c \), they are in continued proportion if the ratio of the first number to the second is equal to the ratio of the second number to the third. Mathematically, this can be expressed as:
\[ \frac{a}{b} = \frac{b}{c} \]

### Step 2: Identify the Numbers
Here, we have:
- \( a = 6 \)
- \( b = 36 \)
- \( c = 216 \)

### Step 3: Set Up the Ratios
Now, we need to set up the ratios:
\[ \frac{6}{36} \quad \text{and} \quad \frac{36}{216} \]

### Step 4: Simplify the First Ratio
Let's simplify the first ratio \( \frac{6}{36} \):
\[ \frac{6}{36} = \frac{1}{6} \]

### Step 5: Simplify the Second Ratio
Now, simplify the second ratio \( \frac{36}{216} \):
\[ \frac{36}{216} = \frac{1}{6} \]

### Step 6: Compare the Ratios
Now we compare the two simplified ratios:
\[ \frac{6}{36} = \frac{1}{6} \quad \text{and} \quad \frac{36}{216} = \frac{1}{6} \]

Since both ratios are equal, we can conclude that:
\[ \frac{6}{36} = \frac{36}{216} \]

### Step 7: Conclusion
Therefore, the numbers 6, 36, and 216 are in continued proportion.

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