n nth term ∆1 ∆2
1 2
10
2 12 2
12
3 24 2
14
4 38 2
16
5 54 2
18
6 72
The first two columns of this table give us n, the position of the term in the series and the term itself. The ∆1 column shows the first difference between consecutive terms, and the ∆2 column shows the second difference, that is, the difference between consecutive differences. The ∆2 values are constant so since this is the second difference column, we know that a quadratic function will give us a formula for calculating the nth term, a_n:
a_n=an²+bn+c where a, b and c are constants. We have three unknown constants so we need three equations to find them, so let’s use the first three terms in the series.
For n=1: 2=a+b+c ➀
For n=2: 12=4a+2b+c ➁
For n=3: 24=9a+3b+c ➂
Now we can solve:
➂-➀: 22=8a+2b, which reduces to 11=4a+b ➃
➁-➀: 10=3a+b ➄
➃-➄: 1=a, so a=1 and from ➄, b=10-3a=10-3=7
c=2-(a+b) from ➀, c=2-8=-6.
Therefore a_n=n²+7n-6.
To check, we put n=4, 5, 6:
a₄=16+28-6=38 OK!
a₅=25+35-6=54 OK!
a₆=36+42-6=72 OK!
The formula checks out.