Of three positive integers, the second is twice the first, and the third is twice the second. One of these integers is 17 more than another. What is the sum of the three integers?

Let's denote the three integers as \( a \), \( b \), and \( c \). According to the problem:

1. The second integer is twice the first:  
   \[
   b = 2a
   \]

2. The third integer is twice the second:  
   \[
   c = 2b
   \]

3. One of these integers is 17 more than another:  
   We need to explore the possible cases here.

### Case 1: \( b = a + 17 \)
Substituting \( b = 2a \):
\[
2a = a + 17
\]
\[
a = 17
\]
Then:
\[
b = 2 \times 17 = 34
\]
\[
c = 2 \times 34 = 68
\]

### Case 2: \( c = b + 17 \)
Substituting \( c = 2b \):
\[
2b = b + 17
\]
\[
b = 17
\]
Then:
\[
a = \frac{b}{2} = \frac{17}{2}
\]
This result is not an integer, so this case is invalid.

### Case 3: \( c = a + 17 \)
Substituting \( c = 2b \) and \( b = 2a \):
\[
2b = a + 17
\]
Substitute \( b = 2a \):
\[
2(2a) = a + 17
\]
\[
4a = a + 17
\]
\[
3a = 17
\]
\[
a = \frac{17}{3}
\]
This result is not an integer, so this case is also invalid.

### Conclusion:
The only valid case is \( a = 17 \), \( b = 34 \), and \( c = 68 \).

### Sum of the three integers:
\[
a + b + c = 17 + 34 + 68 = 119
\]

So, the sum of the three integers is \( \boxed{119} \).

Would you like to explore any details or ask questions? Here are some related problems:

1. If the first integer was 20, what would be the sum of the three integers under the same conditions?
2. What if the second integer was 3 times the first instead of twice? How would that change the solution?
3. Can you derive a general formula for the sum given any multiplier instead of 2?
4. How does this problem change if all integers must be even?
5. Can the sum of the three integers be a prime number under similar conditions?

**Tip:** When dealing with multiple unknowns, try to express them all in terms of one variable to simplify calculations.

Explore the solution to a problem involving three positive integers, where the second is twice the first and the third is twice the second. Understand how to find the sum of these integers and related mathematical concepts. Suitable for students in Grades 7-9.

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