To solve the equation
[tex]\[ a + \frac{1}{b + \frac{1}{c}} = \frac{38}{11}, \][/tex]
where [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] are natural numbers, follow these steps:
1. Isolate the fraction: Start by isolating the fractional part of the equation.
[tex]\[ \frac{1}{b + \frac{1}{c}} = \frac{38}{11} - a. \][/tex]
2. Consider valid values for [tex]\(a\)[/tex]: Since [tex]\(a\)[/tex] is a natural number, it means [tex]\(a\)[/tex] should be chosen such that [tex]\(\frac{38}{11} - a\)[/tex] remains positive and makes the right-hand side a valid fraction.
3. Try different values for [tex]\(a\)[/tex]: Begin checking natural number values for [tex]\(a\)[/tex] starting from 1 up to values that make sense given the left side of the equation.
Let's test [tex]\(a = 3\)[/tex]:
[tex]\[ \frac{1}{b + \frac{1}{c}} = \frac{38}{11} - 3 = \frac{38}{11} - \frac{33}{11} = \frac{5}{11}. \][/tex]
So now, we need to solve:
[tex]\[ \frac{1}{b + \frac{1}{c}} = \frac{5}{11}. \][/tex]
4. Invert both sides:
[tex]\[ b + \frac{1}{c} = \frac{11}{5}. \][/tex]
5. Express [tex]\(\frac{1}{c}\)[/tex] explicitly:
[tex]\[ b + \frac{1}{c} = 2 + \frac{1}{5}. \][/tex]
This implies:
[tex]\[ b = 2 \quad \text{and} \quad \frac{1}{c} = \frac{1}{5}, \][/tex]
which means that [tex]\( c = 5 \)[/tex].
Therefore, a set of values that satisfies the equation is:
[tex]\[ a = 3, \quad b = 2, \quad c = 5. \][/tex]
6. Verify the solution: Substitute [tex]\(a = 3\)[/tex], [tex]\(b = 2\)[/tex], and [tex]\(c = 5\)[/tex] back into the original equation to ensure correctness.
[tex]\[ 3 + \frac{1}{2 + \frac{1}{5}} = 3 + \frac{1}{2 + 0.2} = 3 + \frac{1}{2.2} = 3 + \frac{5}{11} = 3 + 0.4545 = 3.4545 \approx \frac{38}{11}. \][/tex]
This verifies the solution since 3.4545 is approximately 3.4545. Thus, we have the right combination of natural numbers:
The values are [tex]\(a = 3\)[/tex], [tex]\(b = 2\)[/tex], and [tex]\(c = 5\)[/tex].
