Answer:
an = 3n^2 +2n +5
Step-by-step explanation:
If the n-th term is given by ...
an = a·n^2 +bn +c
Then we can write 3 equations in a, b, c:
10 = a·1 +b·1 + c
21 = a·4 +b·2 +c
38 = a·9 +b·3 +c
Subtracting each equation from the next gives ...
11 = 3a +b
17 = 5a +b
Then subtracting the first of these from the second, we have ...
6 = 2a
3 = a
Substituting in to the first of the reduced equations, we find ...
11 = 3(3) +b ⇒ b = 2
And substituting these values into the first equation gives ...
10 = 3 +2 +c ⇒ c = 5
The expression for the n-th term is ...
an = 3n^2 +2n +5
