What is the nth term rule of the quadratic sequence below? 12
,
17
,
24
,
33
,
44
,
57
,
72
,
.
.
.

9514 1404 393

  a[n] = n² +2n +9

Step-by-step explanation:

First differences are ...

  17-12 = 5; 24-17 = 7; 33-24 = 9 . . . . . . define d1=5

So, second differences are ...

  7-5 = 2; 9-7 = 2 . . . . . . . define d2=2

The coefficient of the n² term is half the second difference (=d2/2). The 0-th term will be smaller than the first term by the difference between the first first difference and the second difference (d1-d2):

  a0 = a1 -(d1 -d2) = 12 -5 +2 = 9

So, the quadratic rule will be ...

  a[n] = n² +9 +kn

for some value of k.

We can find k using the first term of the sequence.

  a[1] = 1² +9 +k·1 = 12

  k = 12 -10 = 2

Then the rule is ...

  a[n] = n² +2n +9

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