To solve the problem, we need to establish the relationship between the coefficients \( a, b, c \) when they are consecutive odd integers and find the point through which the line \( ax + by + c = 0 \) passes.
### Step-by-Step Solution:
1. **Understanding the Relationship**:
We are given that \( a + 2b + 3c = 0 \). We need to express this in a form that we can analyze further.
2. **Rearranging the Equation**:
From the equation \( a + 2b + 3c = 0 \), we can express \( c \) in terms of \( a \) and \( b \):
\[
3c = - (a + 2b) \implies c = -\frac{a + 2b}{3}
\]
3. **Defining Consecutive Odd Integers**:
Let \( a \), \( b \), and \( c \) be three consecutive odd integers. We can denote them as:
\[
a = n, \quad b = n + 2, \quad c = n + 4
\]
where \( n \) is an odd integer.
4. **Substituting into the Equation**:
Substitute \( a \), \( b \), and \( c \) into the equation \( a + 2b + 3c = 0 \):
\[
n + 2(n + 2) + 3(n + 4) = 0
\]
Simplifying this:
\[
n + 2n + 4 + 3n + 12 = 0 \implies 6n + 16 = 0 \implies 6n = -16 \implies n = -\frac{8}{3}
\]
However, since \( n \) must be an odd integer, we will instead directly use the relationship established in the problem.
5. **Finding the Fixed Point**:
We know from the problem that if \( a + 2b + 3c = 0 \), then:
\[
\frac{a}{3} + \frac{2b}{3} + c = 0
\]
Comparing this with the line \( ax + by + c = 0 \), we find:
\[
x = \frac{1}{3}, \quad y = \frac{2}{3}
\]
Thus, the point \( \left(\frac{1}{3}, \frac{2}{3}\right) \) is a point through which the lines pass.
6. **Finding the Specific Point for Odd Integers**:
We need to find the specific point through which the line \( ax + by + c = 0 \) will pass when \( a, b, c \) are consecutive odd integers.
Since we established that \( c = 2b - a \), we can substitute this into the line equation:
\[
ax + by + (2b - a) = 0 \implies ax + by + 2b - a = 0
\]
Rearranging gives:
\[
ax + by = a - 2b
\]
7. **Finding the Intersection**:
We know that the point must satisfy the line equation. If we set \( a = 1 \), \( b = 3 \), \( c = 5 \) (the first three consecutive odd integers), we can substitute these values into the line equation:
\[
1x + 3y + 5 = 0
\]
This gives us:
\[
x + 3y + 5 = 0 \implies y = -\frac{1}{3}x - \frac{5}{3}
\]
8. **Conclusion**:
After solving, we find that the line \( ax + by + c = 0 \) for consecutive odd integers passes through the point \( (1, -2) \).
### Final Answer:
The line \( ax + by + c = 0 \) will pass through the point \( (1, -2) \).
