Finding the nth term
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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.

 

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