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 $n$?
1. Let the number of pages in the first volume be \( x \).
2. The number of pages in the second volume is \( x + 50 \).
3. The number of pages in the third volume is \( \frac{3}{2}(x + 50) \).

4. The first page of the first volume is page number 1.
5. The first page of the second volume is page number \( x + 1 \).
6. The first page of the third volume is page number \( x + (x + 50) + 1 = 2x + 51 \).

7. According to the problem, the sum of the page numbers on the first pages of the three volumes is 1709:
   \[
   1 + (x + 1) + (2x + 51) = 1709
   \]

8. Simplify the equation:
   \[
   1 + x + 1 + 2x + 51 = 1709
   \]
   \[
   3x + 53 = 1709
   \]

9. Solve for \( x \):
   \[
   3x = 1709 - 53
   \]
   \[
   3x = 1656
   \]
   \[
   x = \frac{1656}{3}
   \]
   \[
   x = 552
   \]

10. The number of pages in the first volume is \( x = 552 \).
11. The number of pages in the second volume is \( x + 50 = 552 + 50 = 602 \).
12. The number of pages in the third volume is \( \frac{3}{2}(x + 50) = \frac{3}{2}(552 + 50) = \frac{3}{2} \times 602 = 903 \).

13. The last page number \( n \) is the sum of the pages in all three volumes:
    \[
    n = 552 + 602 + 903 = 2057
    \]

14. To find the largest prime factor of 2057, we perform prime factorization:
    - Check divisibility by 2: 2057 is odd, not divisible by 2.
    - Check divisibility by 3: Sum of digits \( 2 + 0 + 5 + 7 = 14 \), not divisible by 3.
    - Check divisibility by 5: Last digit is not 0 or 5.
    - Check divisibility by 7: \( 2057 \div 7 \approx 293.857 \), not divisible by 7.
    - Check divisibility by 11: Alternating sum of digits \( 2 - 0 + 5 - 7 = 0 \), divisible by 11.
    - \( 2057 \div 11 = 187 \).

15. Factorize 187:
    - Check divisibility by 2: 187 is odd, not divisible by 2.
    - Check divisibility by 3: Sum of digits \( 1 + 8 + 7 = 16 \), not divisible by 3.
    - Check divisibility by 5: Last digit is not 0 or 5.
    - Check divisibility by 7: \( 187 \div 7 \approx 26.714 \), not divisible by 7.
    - Check divisibility by 11: \( 187 \div 11 \approx 17 \), divisible by 11.
    - \( 187 = 11 \times 17 \).

16. The prime factors of 2057 are 11 and 17. The largest prime factor is 17.

The final answer is \(\boxed{17}\)