Pictures of sequences

Your book's Table 2 draws number sequences as pictures made of dots. A picture shows why a sequence has its name.

Triangular numbers 1, 3, 6, 10, 15: that many dots make a triangle. Put 1 dot in the top row, 2 under it, then 3, 4, 5. Rows of 1, 2, 3 and 4 make 1 + 2 + 3 + 4 = 10 dots. The next triangle just adds a new bottom row, one dot longer. (With 4 or 5 dots you cannot make such a neat triangle.)

Squares 1, 4, 9, 16, 25: that many dots make a square grid: 2 rows of 2, 3 rows of 3, 4 rows of 4.

Cubes 1, 8, 27, 64, 125: that many little cubes build a big cube. Stack 8 dice and you get a bigger cube, 2 along each edge. Could 9 dice make a cube? No: the next cube needs 3 × 3 × 3 = 27.

One number can play two roles. 36 dots make a square, 6 rows of 6. They also make a triangle with rows of 1, 2, 3, 4, 5, 6, 7 and 8, because 1 + 2 + … + 8 = 36. So 36 is both a square number and a triangular number.

Hexagonal numbers 1, 7, 19, 37, …: start with 1 dot, then put a ring of 6 dots round it to make a hexagon of 7. Each new ring holds 6 more dots than the last: 6, 12, 18, 24. So after 37 comes 37 + 24 = 61.