Rows in Pascal's Triangle containing Numbers in Arithmetic Sequence

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Theorem

There are an infinite number of rows of Pascal's triangle which contain $3$ integers in arithmetic sequence.


Proof

Suppose $\dbinom n k$, $\dbinom n {k + 1}$ and $\dbinom n {k + 2}$ are in an arithmetic sequence.

Then:

\(\ds \dbinom n {k + 2} - \dbinom n {k + 1}\) \(=\) \(\ds \dbinom n {k + 1} - \dbinom n k\) Definition of Arithmetic Sequence
\(\ds \frac {n!} {\paren {n - k - 2}! \paren {k + 2}!} - \frac {n!} {\paren {n - k - 1}! \paren {k + 1}!}\) \(=\) \(\ds \frac {n!} {\paren {n - k - 1}! \paren {k + 1}!} - \frac {n!} {\paren {n - k}! \paren k!}\) Definition of Binomial Coefficient
\(\ds \paren {n - k - 1} \paren {n - k} - \paren {n - k} \paren {k + 2}\) \(=\) \(\ds \paren {n - k} \paren {k + 2} - \paren {k + 1} \paren {k + 2}\) Multiply both sides by $\dfrac {\paren {n - k}! \paren {k + 2}!} {n!}$
\(\ds n^2 - k n - k n + k^2 - n + k - n k - 2 n + k^2 + 2 k\) \(=\) \(\ds n k + 2 n - k^2 - 2 k - k^2 - 2 k - k - 2\)
\(\ds n^2 - \paren {4 k + 5} n + \paren {4 k^2 + 8 k + 2}\) \(=\) \(\ds 0\)
\(\ds n\) \(=\) \(\ds \frac {4 k + 5 \pm \sqrt {\paren {4 k + 5}^2 - 4 \paren {4 k^2 + 8 k + 2} } } 2\) Quadratic Formula
\(\ds \) \(=\) \(\ds \frac {4 k + 5 \pm \sqrt {16 k^2 + 40 k + 25 - 16 k^2 - 32 k - 8} } 2\)
\(\ds \) \(=\) \(\ds \frac {4 k + 5 \pm \sqrt {8 k + 17} } 2\)

Since $n$ is rational, we require $8 k + 17$ to be a square.

Since $8 k + 17$ is odd, if $8 k + 17$ is square, then $\sqrt {8 k + 17}$ is odd.

Write $\sqrt {8 k + 17} = 2 x + 1$.

Notice that:

\(\ds \paren {2 x + 1}^2\) \(=\) \(\ds 4 x^2 + 4 x + 1\)
\(\ds \) \(=\) \(\ds 8 \paren {\frac {x^2 + x - 4} 2} + 17\)

Using the substitution $k = \dfrac {x^2 + x - 4} 2$:

\(\ds n\) \(=\) \(\ds \frac {2 x^2 + 2 x - 8 + 5 \pm \paren {2 x + 1} } 2\)
\(\ds \) \(=\) \(\ds x^2 - 2 \text { or } x^2 + 2 x - 1\)
\(\ds \) \(=\) \(\ds x^2 - 2 \text { or } \paren {x + 1}^2 - 2\)

Each $x$ with $k = \dfrac {x^2 + x - 4} 2 > 0$ give a value for $n$.

Therefore there are an infinite number of rows of Pascal's triangle which contain $3$ integers in arithmetic sequence.

$\blacksquare$


Examples

$7, 21, 35$

The integers:

$7, 21, 35$

are in arithmetic sequence and appear in row $7$ of Pascal's triangle.


$1001, 2002, 3003$

The integers:

$1001, 2002, 3003$

are in arithmetic sequence and appear in row $14$ of Pascal's triangle.


$490 \,314, 817 \, 190, 1 \, 144 \, 066$

The integers:

$490 \, 314, 817 \, 190, 1 \, 144 \, 066$

are in arithmetic sequence and appear in row $23$ of Pascal's triangle.


Also see


Historical Note

This result, along with Rows in Pascal's Triangle containing Numbers in Geometric Sequence and Rows in Pascal's Triangle containing Numbers in Harmonic Sequence, was apparently published by Theodore Samuel Motzkin in Volume $12$ of Scripta Mathematica, but details have not been established.


Sources

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