Second-Order Sequences

= + +

Tap a term to see what it means, or .

Calculate

Example: snooker balls racked in a triangle — 1, 3, 6, 10… How many balls are in a triangle with 100 rows? Change the first terms and the position to work out your own case.

First terms At least three, separated by spaces or commas. Position n

Step by step

What each term means

General Term (aₙ): The term at position n, computed directly, without listing the earlier ones.
First Term (a₁): Where the sequence starts.
First Difference (Δ₁): a₂ − a₁, the first jump. In an arithmetic sequence, the jumps are all equal and the formula would stop here.
Second Difference (Δ₂) and Its Weight: How much each jump grows over the previous one. Up to position n, it builds up (n − 1)(n − 2)/2 times — a triangular number.