Given [tex]\(a, b, c\)[/tex] are natural numbers, solve for [tex]\(a, b,\)[/tex] and [tex]\(c\)[/tex]:

[tex]\[ a + \frac{1}{b + \frac{1}{c}} = \frac{38}{11} \][/tex]



Answer :

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].