Find a quadratic model for the sequence with the indicated terms.
$$a _ { 0 } = 7 , a _ { 2 } = 3 , a _ { 5 } = 12$$
A) $a _ { n } = n ^ { 2 } + 7$
B) $a _ { n } = n ^ { 2 } - 4 n + 7$
C) $a _ { n } = n ^ { 2 } - 7 n + 7$
D) $a _ { n } = n ^ { 2 } - 7 n + 4$
E) $a _ { n } = n ^ { 2 } - n + 7$

Question

Quizplus · Accepted Answer

To find the correct quadratic model, we substitute the given terms into the general form of a quadratic sequence, $a_n = an^2 + bn + c$, and solve for $a$, $b$, and $c$.Given $a_0 = 7$, $a_2 = 3$, and $a_5 = 12$, we have:1. For $n = 0$, $a_0 = c = 7$.2. For $n = 2$, $a_2 = 4a + 2b + c = 3$. Substituting $c = 7$, we get $4a + 2b = -4$.3. For $n = 5$, $a_5 = 25a + 5b + c = 12$. Substituting $c = 7$, we get $25a + 5b = 5$.Solving the system of equations for $a$ and $b$, we find $a = 1$ and $b = -4$. Thus, the quadratic model is $a_n = n^2 - 4n + 7$, which matches choice B.

Find a Quadratic Model for the Sequence with the Indicated a0=7,a2=3,a5=12a _ { 0 } = 7 , a _ { 2 } = 3 , a _ { 5 } = 12a0​=7,a2​=3,a5​=12

Find a Quadratic Model for the Sequence with the Indicated $a _ { 0 } = 7 , a _ { 2 } = 3 , a _ { 5 } = 12$