If a,b and c are three natural numbers in ascending order, then

Options

A. `c ^(2) - a ^(2) = b ^(2)`
B. `c ^(2) - a^(2) lt b ^(2)`
C. `c ^(2) +b ^(2) = a ^(2)`
D. `c ^(2) - a ^(2) gt b `

Answer

To solve the problem, we need to analyze the relationship between three natural numbers \( a \), \( b \), and \( c \) that are in ascending order. We will verify the property that the difference of the square of the third number and the first number is greater than the square of the second number.

### Step-by-Step Solution:

1. **Understanding the Order of Numbers**:
Since \( a \), \( b \), and \( c \) are in ascending order, we can say:
\[ a < b < c \]
This means \( a \) is the smallest, \( b \) is the middle number, and \( c \) is the largest.

2. **Setting Up the Expression**:
We need to evaluate the expression:
\[ c^2 - a^2 \]
and compare it with:
\[ b^2 \]

3. **Using the Difference of Squares**:
We can use the difference of squares formula:
\[ c^2 - a^2 = (c - a)(c + a) \]
This shows that the difference of the squares can be expressed in terms of the sum and difference of \( c \) and \( a \).

4. **Analyzing the Terms**:
Since \( c > a \), both \( c - a \) and \( c + a \) are positive. Therefore, \( c^2 - a^2 \) is positive.

5. **Comparing with \( b^2 \)**:
We need to show that:
\[ c^2 - a^2 > b^2 \]
Rearranging gives:
\[ c^2 - b^2 > a^2 \]
This can be rewritten using the difference of squares again:
\[ (c - b)(c + b) > a^2 \]
Since \( c > b \) and \( c + b > 0 \), the left-hand side is positive.

6. **Choosing Specific Values**:
To verify, let’s choose specific natural numbers:
- Let \( a = 1 \), \( b = 2 \), and \( c = 3 \).
- Calculate \( c^2 - a^2 \):
\[ c^2 - a^2 = 3^2 - 1^2 = 9 - 1 = 8 \]
- Calculate \( b^2 \):
\[ b^2 = 2^2 = 4 \]
- Now compare:
\[ 8 > 4 \]
This confirms that \( c^2 - a^2 > b^2 \).

7. **Conclusion**:
The property holds true for the chosen values, and we can generalize that for any three natural numbers \( a < b < c \):
\[ c^2 - a^2 > b^2 \]

### Final Answer:
The correct option is that \( c^2 - a^2 > b^2 \).

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