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# AP Train

**Forming an equation using the amounts added and subtracted**

To shift the sequence along $D$ spaces,

from $(1111)+ (1112)+( 1113)+, \dots, (1111 + N-1)$

to $(1111+D)+(1112+D) + (1113+D) + \dots+(1111 + N -1+ D),$

you add $D$ to each term, so you add it $N$ times, since there are $N$ terms.

That is the same as adding a total of $N\times D$.

Then removing the first term, $1111+D$, takes the sum back to its original value.

So $N\times D$ must be equal to $1111+D$.

$$\begin{align}ND&=1111+D\\\Rightarrow ND-D&=1111\\\Rightarrow (N-1)D&=1111\end{align}$$

$N$ and $D$ are both integers, so $N-1$ and $D$ must be factors of $1111$.

$1111=11\times101$, which are both prime, so all of the possible values of $N$ and $D$ are shown below:

You can see some of these sums in action on the spreadsheet screenshot at the bottom of the page.

**Using the formula for the sum of the first $n$ natural numbers**

The sum of the first $n$ natural numbers is equal to $$

1+2+\dots + (n-1) + n =\frac{1}{2}n(n+1)

$$

Its easier in a problem like this to introduce some notation. Let's write $S(n)$ to mean the sum of the first $n$ natural numbers.

Since the two sums are equal we have

$$

S(1111+N-1)-S(1110) = S(1111+N -1 +D) - S(1111 + D)

$$

Let's put the formula into each of these and cancel each factor of a half.

$$

(1110+N)(1111+N)-(1110)(1111)=(1110+N+D)(1111+N+D)-(1111+D)(1112+D)

$$

Before leaping in we can see that many parts cancel, so we can put

$$

2221N + N^2 = 2221(N+D) + (N+D)^2 - 2223D - D^2- 2222

$$

Collecting things together gives

$$

D(N-1) = 1111

$$

Now for a bit of number theory: $1111 = 11\times 101$, or $1\times1111$. Thus, for solutions we require:

$D = 11, N=102$ or $D=101, N=12$ or $D=1,N=1112$ or $D=1111,N=2$.

You can see some of these sums in action on this spreadsheet screenshot

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Age 16 to 18

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To shift the sequence along $D$ spaces,

from $(1111)+ (1112)+( 1113)+, \dots, (1111 + N-1)$

to $(1111+D)+(1112+D) + (1113+D) + \dots+(1111 + N -1+ D),$

you add $D$ to each term, so you add it $N$ times, since there are $N$ terms.

That is the same as adding a total of $N\times D$.

Then removing the first term, $1111+D$, takes the sum back to its original value.

So $N\times D$ must be equal to $1111+D$.

$$\begin{align}ND&=1111+D\\\Rightarrow ND-D&=1111\\\Rightarrow (N-1)D&=1111\end{align}$$

$N$ and $D$ are both integers, so $N-1$ and $D$ must be factors of $1111$.

$1111=11\times101$, which are both prime, so all of the possible values of $N$ and $D$ are shown below:

Product $(N-1)\times D =1111$ | $N$ and $D$ |
---|---|

$1111\times1$ | $N=1112,D=1$ |

$1\times1111$ | $N=2,D=1111$ |

$11\times101$ | $N=12,D=101$ |

$101\times11$ | $N=102,D=11$ |

You can see some of these sums in action on the spreadsheet screenshot at the bottom of the page.

The sum of the first $n$ natural numbers is equal to $$

1+2+\dots + (n-1) + n =\frac{1}{2}n(n+1)

$$

Its easier in a problem like this to introduce some notation. Let's write $S(n)$ to mean the sum of the first $n$ natural numbers.

Since the two sums are equal we have

$$

S(1111+N-1)-S(1110) = S(1111+N -1 +D) - S(1111 + D)

$$

Let's put the formula into each of these and cancel each factor of a half.

$$

(1110+N)(1111+N)-(1110)(1111)=(1110+N+D)(1111+N+D)-(1111+D)(1112+D)

$$

Before leaping in we can see that many parts cancel, so we can put

$$

2221N + N^2 = 2221(N+D) + (N+D)^2 - 2223D - D^2- 2222

$$

Collecting things together gives

$$

D(N-1) = 1111

$$

Now for a bit of number theory: $1111 = 11\times 101$, or $1\times1111$. Thus, for solutions we require:

$D = 11, N=102$ or $D=101, N=12$ or $D=1,N=1112$ or $D=1111,N=2$.

You can see some of these sums in action on this spreadsheet screenshot