What's the recursive formula? -1, 1, 4, 8, 13

1 Answer
Mar 23, 2018

x_1=-1

x_(n+1)=x_n+n+1

Explanation:

The first term, x_1, is -1.

Watch what happens when we take the first difference between consecutive terms.

When n=1 , x_(n+1)=x_(1+1)=x_2.

x_(n+1)-x_n=x_2-x_1=1-(-1) =2=n+1

When n=2 , x_(n+1)=x_(2+1)=x_3.

x_(n+1)-x_n=x_3-x_2=4-1=3=n+1

When n=3 , x_(n+1)=x_(3+1)=x_4.

x_(n+1)-x_n=x_4-x_3=8-4=4=n+1

When n=4 , x_(n+1)=x_(4+1)=x_5.

x_(n+1)-x_n=x_5-x_4=13-8=5=n+1

Note that in each case, x_(n+1)-x_n=n+1.

Adding x_n to both sides of this equation gives

x_(n+1)=x_n+n+1.