Suppose the second term is x and the third term y
sequence to 3 terms: 27 x y
First differences x-27 y-x
Second difference 2
The first of the first differences is 1.
That means x-27=1
x=28
...constant second differences of 2...
That means (y-x)-(x-27) = 2
y-x-x+27 = 2
y-2x+27 = 2
y-2x = -25
Substitute x=28
y = 2(28)-25
y = 56-25
y = 31
So the difference table above becomes:
sequence to 3 terms: 27 28 31
First differences 1 3
Second difference 2
To extend the table to 2 more terms put 2 more 2's on the line with
the secnd differences, and put letters A and B for the next two terms
and C and D for the next two first differences.
sequence to 5 terms: 27 28 31 A B
First differences 1 3 C D
Second difference 2 2 2
Now we work backwards:
C must be 5 so that C-3 will equal 2
sequence to 5 terms: 27 28 31 A B
First differences 1 3 5 D
Second difference 2 2 2
D must be 7 so that D-5 will equal 2
sequence to 5 terms: 27 28 31 A B
First differences 1 3 5 7
Second difference 2 2 2
A must be 36 so that A-31 will equal 5
sequence to 5 terms: 27 28 31 36 B
First differences 1 3 5 7
Second difference 2 2 2
Finally B must be 43 so that B-36 will equal 7
sequence to 5 terms: 27 28 31 36 43
First differences 1 3 5 7
Second difference 2 2 2
So the next two terms are 36 and 43
Edwin