Precalculus, Chapter 9, 9.4, Section 9.4, Problem 70

You need to remember what a quadratic model is, such that:
a_n = f(n) = a*n^2 + b*n + c
The problem provides the following information, such that:
a_0 = 7 => f(0) = a*0^2 + b*0 + c => c = 7
a_1 = 6 => f(1) = a*1^2 + b*1 + c => a + b + c =6
a_3 = 10 => f(3) = a*3^2 + b*3 + c => 9a + 3b + c = 10
You need to replace 7 for c in equation a + b + c = 6 :
a + b +7 =6=> a + b = -1
You need to replace 7 for c in equation 9a + 3b + c = 10:
9a + 3b +7 = 10=> 9a + 3b = 3 => 3a + b = 1
Subtract a + b =-1 from 3a + b = 1 , such that:
3a + b - a - b= 1 + 1
2a = 2=> a = 1
Replace 1 for a in equation a + b =-1 such that:
1+ b = -1 => b = -2
Hence, the quadratic model for the given sequence is a_n = n^2 - 2n + 7.