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# AP Train

A particular list of $N$ consecutive integers starts with $1111$ as follows:

$$1111, 1112, 1113, \dots, 1111 + N-1$$

The entire list is shifted $D$ places along the number line and the first number then excluded, leaving a list of $N-1$ larger consecutive numbers as follows:

$$

1112+D, 1113+D, \dots, 1111 + N -1+ D

$$

In each list the sum of the integers is the same.

What are the possibilities for $N$ and $D$?

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A particular list of $N$ consecutive integers starts with $1111$ as follows:

$$1111, 1112, 1113, \dots, 1111 + N-1$$

The entire list is shifted $D$ places along the number line and the first number then excluded, leaving a list of $N-1$ larger consecutive numbers as follows:

$$

1112+D, 1113+D, \dots, 1111 + N -1+ D

$$

In each list the sum of the integers is the same.

What are the possibilities for $N$ and $D$?

Did you know ... ?

Progressions of integers occur remarkably frequently in mathematics in applications from quantum mechanics to number theory and they have many beautiful properties. Even Carl Gauss, possibly the greatest mathematician of all time, fondly recalled his first encounter with sums of consecutive natural numbers, when he noticed that the sum of the first $100$ whole numbers equalled $50$ lots of $101$.

Progressions of integers occur remarkably frequently in mathematics in applications from quantum mechanics to number theory and they have many beautiful properties. Even Carl Gauss, possibly the greatest mathematician of all time, fondly recalled his first encounter with sums of consecutive natural numbers, when he noticed that the sum of the first $100$ whole numbers equalled $50$ lots of $101$.