Adding up a sequence

A new way to make one sequence from another: keep a running total. Take the first number, then the first two added, then the first three, and so on. Each total is a number of the new sequence.

Start with the all 1's: 1 = 1, 1 + 1 = 2, 1 + 1 + 1 = 3, 1 + 1 + 1 + 1 = 4. The totals are 1, 2, 3, 4, …: adding up the all 1's gives the counting numbers.

Now add up the counting numbers: 1 = 1, 1 + 2 = 3, 1 + 2 + 3 = 6, 1 + 2 + 3 + 4 = 10, 1 + 2 + 3 + 4 + 5 = 15. The totals 1, 3, 6, 10, 15, … are the triangular numbers! The picture shows why: rows of 1, 2, 3 and 4 dots make a triangle of 10 dots.

A shortcut for a triangular number: multiply the number of rows by one more than that, then halve it. 5 rows: 5 × 6 ÷ 2 = 15. 7 rows: 7 × 8 ÷ 2 = 28.

Now add two triangular numbers that sit next to each other: 1 + 3 = 4, 3 + 6 = 9, 6 + 10 = 16, 10 + 15 = 25. These are the square numbers! Take a square of 4 rows of 4 dots and cut it along a slanting line from corner to corner, keeping the dots on the line on one side. One part has rows of 1, 2, 3, 4 dots (10); the other has rows of 1, 2, 3 (6). So 6 + 10 = 16 = 4 × 4.