Is there an easier way to work out a number sequence which isn't linear?
If we know that the formula is a quadratic, there is a sneaky way,
. . but you may need some background.
We begin with a basic quadratic function: .
This generates the Triangular Numbers: 1, 3, 6, 10, 15, . . .
And it should be obvious why they are called "triangular."
o o o
o o o o o o
o o o o o o o o o o
1 3 6 10
So we have the formula: .
. . and we will modify it to fit the given sequence: 1, 2, 4, 7, 11, 16, 22, . . .
For the first few values, compare our sequence to the triangular numbers
The values of do not match those of
. . Some adjustments are necessary . . . let's examine them.
We see that we must subtract a number one less than
. . Hence: .
And we have: .
. . Therefore: .