To examine the Pythagorean triples presented for different [tex]\(x\)[/tex]-values and identify any patterns, let's look at the differences between the three values [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] in each triple.
Here are the given triples again for reference:
[tex]\[
\begin{array}{|c|c|}
\hline
x\text{-value} & \text{Pythagorean Triple} \\
\hline
7 & (14, 48, 50) \\
\hline
8 & (16, 63, 65) \\
\hline
9 & (18, 80, 82) \\
\hline
10 & (20, 99, 101) \\
\hline
\end{array}
\][/tex]
### Analyzing the Patterns
Let's look at each triple and examine the differences between [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
1. For [tex]\(x = 7\)[/tex]:
[tex]\[
(a, b, c) = (14, 48, 50)
\][/tex]
Differences:
[tex]\[
b - a = 48 - 14 = 34
\][/tex]
[tex]\[
c - b = 50 - 48 = 2
\][/tex]
2. For [tex]\(x = 8\)[/tex]:
[tex]\[
(a, b, c) = (16, 63, 65)
\][/tex]
Differences:
[tex]\[
b - a = 63 - 16 = 47
\][/tex]
[tex]\[
c - b = 65 - 63 = 2
\][/tex]
3. For [tex]\(x = 9\)[/tex]:
[tex]\[
(a, b, c) = (18, 80, 82)
\][/tex]
Differences:
[tex]\[
b - a = 80 - 18 = 62
\][/tex]
[tex]\[
c - b = 82 - 80 = 2
\][/tex]
4. For [tex]\(x = 10\)[/tex]:
[tex]\[
(a, b, c) = (20, 99, 101)
\][/tex]
Differences:
[tex]\[
b - a = 99 - 20 = 79
\][/tex]
[tex]\[
c - b = 101 - 99 = 2
\][/tex]
### Pattern Observed
From the differences calculated above, we observe the following patterns among the given Pythagorean triples:
- The difference [tex]\(b - a\)[/tex] increases as [tex]\(x\)[/tex] increases.
- The difference [tex]\(c - b\)[/tex] is consistently [tex]\(2\)[/tex] for all the triples.
### Generating Triples Using the Identity
The identity used to generate the triples is:
[tex]\[
(a, b, c) = (2x, x^2 - 1, x^2 + 1)
\][/tex]
This consistently produces triples where:
[tex]\[
c - b = 2
\][/tex]
### Example of a Pythagorean Triple Not Fitting This Pattern
Let's consider a well-known Pythagorean triple that does not follow this pattern and cannot be generated using the given identity:
The triple [tex]\((3, 4, 5)\)[/tex]:
Here:
[tex]\[
b - a = 4 - 3 = 1
\][/tex]
[tex]\[
c - b = 5 - 4 = 1
\][/tex]
This triple [tex]\((3, 4, 5)\)[/tex] does not fit the pattern [tex]\(c - b = 2\)[/tex] and cannot be generated using the identity [tex]\((2x, x^2 - 1, x^2 + 1)\)[/tex], since the identity would produce distinct differences as observed from the given set of triples.
### Conclusion
Upon examining the given set of Pythagorean triples, we see a consistent pattern of the difference [tex]\(c - b = 2\)[/tex]. The classic Pythagorean triple [tex]\((3, 4, 5)\)[/tex], however, does not fit this pattern and cannot be generated using the identity [tex]\( (2x, x^2 - 1, x^2 + 1) \)[/tex].
