A book is published in three volumes, the pages being numbered from 1 onwards. The page numbers are continued from the first volume to the second volume to the third. The number of pages in the second volume is 50 more than that in the first volume, and the number pages in the third volume is one and a half times that in the second. The sum of the page numbers on the first pages of the three volumes is 1709. If n is the last page number, what is the largest prime factor of n ?

Answer

To solve the problem step-by-step, we will define the number of pages in each volume and use the information given to find the last page number \( n \) and then determine the largest prime factor of \( n \).

### Step 1: Define Variables
Let:
- \( x \) = number of pages in the first volume.
- \( y \) = number of pages in the second volume.
- \( z \) = number of pages in the third volume.

### Step 2: Set Up Equations
From the problem statement, we know:
1. The number of pages in the second volume is 50 more than that in the first volume:
\[ y = x + 50 \]
2. The number of pages in the third volume is one and a half times that in the second volume:
\[ z = \frac{3}{2}y = \frac{3}{2}(x + 50) \]

### Step 3: First Page Numbers
The first page numbers of each volume are:
- First volume starts at 1.
- Second volume starts at \( x + 1 \) (since it follows the first volume).
- Third volume starts at \( x + y + 1 = x + (x + 50) + 1 = 2x + 51 \).

### Step 4: Sum of First Page Numbers
According to the problem, the sum of the first pages of the three volumes is 1709:
\[ 1 + (x + 1) + (2x + 51) = 1709 \]
Simplifying this:
\[ 1 + x + 1 + 2x + 51 = 1709 \]
\[ 3x + 53 = 1709 \]
\[ 3x = 1709 - 53 \]
\[ 3x = 1656 \]
\[ x = \frac{1656}{3} = 552 \]

### Step 5: Calculate the Number of Pages in Each Volume
Now we can find \( y \) and \( z \):
1. For the second volume:
\[ y = x + 50 = 552 + 50 = 602 \]
2. For the third volume:
\[ z = \frac{3}{2}(y) = \frac{3}{2}(602) = 903 \]

### Step 6: Calculate the Last Page Number \( n \)
The last page number \( n \) is the total number of pages in all three volumes:
\[ n = x + y + z = 552 + 602 + 903 \]
Calculating this:
\[ n = 552 + 602 = 1154 \]
\[ n = 1154 + 903 = 2057 \]

### Step 7: Find the Largest Prime Factor of \( n \)
Now we need to find the prime factors of \( 2057 \):
1. Check divisibility:
- \( 2057 \) is odd, not divisible by \( 2 \).
- Sum of digits \( 2 + 0 + 5 + 7 = 14 \), not divisible by \( 3 \).
- Last digit is \( 7 \), not divisible by \( 5 \).
- Check \( 7 \): \( 2057 \div 7 \approx 293.857 \) (not divisible).
- Check \( 11 \): \( 2057 \div 11 = 187 \) (exactly divisible).
- Check \( 187 \): \( 187 = 11 \times 17 \).

Thus, the prime factorization of \( 2057 \) is:
\[ 2057 = 11 \times 11 \times 17 \]

### Conclusion
The largest prime factor of \( n \) is:
\[ \boxed{17} \]