9514 1404 393
Answer:
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
