What is the rule for this pattern 2 6 15 31 56 92?

Answer 1

The pattern at each stage is the addition of a square number of increasing value to the previous term.

1 + 12 = 2

2 + 22 = 6

6 + 32 = 15

15 + 42 = 31

31 + 52 = 56

56 + 62 = 92

The formula for the nth term is, a(n) = n3/3 + n2/2 + n/6 + 1

or it can be written a(n) = 1/6(2n3 + 3n2 + n + 6)

So the 7th term = 1/6(686 + 147 + 7 + 6) = 1/6 x 846 = 141 ( = 92 + 72)

NOTE : The rule is the same as the sum of the squared numbers plus one.

So another way of presenting the formula is, a(n) = 1/6[n(n + 1)(2n + 1)] + 1