Rows in Pascal's Triangle containing Numbers in Arithmetic Sequence/Examples/1001, 2002, 3003

Example of Rows in Pascal's Triangle containing Numbers in Arithmetic Sequence

The integers:

$1001, 2002, 3003$

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

Proof

We have:

\(\ds 2002 - 1001\)

\(=\)

\(\ds 1001\)

\(\ds 3003 - 2002\)

\(=\)

\(\ds 1001\)

thus demonstrating the common difference of $1001$.

Then we have that row $14$ of Pascal's triangle is:

$1, 14, 91, 364, \color {red} {1001, 2002, 3003}, 3432, 3003, 2002, 1001, 364, 91, 14, 1$

This sequence is A034870 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

$\blacksquare$

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $35$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $35$