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

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Answer 1 · 2018-03-23T14:49:25

#x_1=-1#

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

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#.