Rows in Pascal's Triangle containing Numbers in Geometric Sequence

Theorem
There exist no rows of Pascal's triangle which contain $3$ integers in geometric sequence.

Proof
Suppose $\dbinom n k$, $\dbinom n {k + 1}$ and $\dbinom n {k + 2}$ are in a geometric sequence.

Then:

Since $n \ge 0$, no row of Pascal's triangle contains $3$ integers in geometric sequence.

However, suppose one extends the definition of binomial coefficients to allow $n < 0$.

Then by Negated Upper Index of Binomial Coefficient, we have:


 * $\dbinom {-1} k = \paren {-1}^k$

which indeed forms a geometric sequence.

Also see

 * Rows in Pascal's Triangle containing Numbers in Arithmetic Sequence
 * Rows in Pascal's Triangle containing Numbers in Harmonic Sequence