c From Part B we know that the sequence can be modeled by a quadratic function,

f (n) = a n^{2} + b n + c,

for some values of

a,

b,

and

c .

To determine the values of

a,

b,

and

c,

let's substitute

1,

2,

and

3

for

n

and use that

f (1) = 3,

f (2) = 10,

f (3) = 21 .

f (n) = a n^{2} + b n + c ⇓ ⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ 3 = a (1)^{2} + b (1) + c 10 = a (2)^{2} + b (2) + c 21 = a (3)^{2} + b (3) + c (I) (II) (III)

Let's use the Substitution Method to find a solution to this system.

⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ 3 = a (1)^{2} + b (1) + c 10 = a (2)^{2} + b (2) + c 21 = a (3)^{2} + b (3) + c (I) (II) (III)

▼

Solve by substitution

$(I), (II), (III):$ Calculate power and product

⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ 3 = a + b + c 10 = 4 a + 2 b + c 21 = 9 a + 3 b + c (I) (II) (III)

$(I), (II), (III):$ Rearrange equation

⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ a + b + c = 3 4 a + 2 b + c = 10 9 a + 3 b + c = 21 (I) (II) (III)

$(I):$ $LHS - a = RHS - a$

⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ b + c = 3 - a 4 a + 2 b + c = 10 9 a + 3 b + c = 21 (I) (II) (III)

$(I):$ $LHS - b = RHS - b$

⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ c = 3 - a - b 4 a + 2 b + c = 10 9 a + 3 b + c = 21 (I) (II) (III)

$(I), (II):$ $c = 3 - a - b$

⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ c = 3 - a - b 4 a + 2 b + 3 - a - b = 10 9 a + 3 b + 3 - a - b = 21 (I) (II) (III)

$(II), (III):$ Add and subtract terms

⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ c = 3 - a - b 3 a + b + 3 = 10 8 a + 2 b + 3 = 21 (I) (II) (III)

$(II), (III):$ $LHS - 3 = RHS - 3$

⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ c = 3 - a - b 3 a + b = 7 8 a + 2 b = 18 (I) (II) (III)

$(II):$ $LHS - 3 a = RHS - 3 a$

⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ c = 3 - a - b b = 7 - 3 a 8 a + 2 b = 18 (I) (II) (III)

$(III):$ $b = 7 - 3 a$

⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ c = 3 - a - b b = 7 - 3 a 8 a + 2 (7 - 3 a) = 18 (I) (II) (III)

$(III):$ Distribute $2$

⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ c = 3 - a - b b = 7 - 3 a 8 a + 14 - 6 a = 18 (I) (II) (III)

$(III):$ Subtract terms

⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ c = 3 - a - b b = 7 - 3 a 2 a + 14 = 18 (I) (II) (III)

$(III):$ $LHS - 14 = RHS - 14$

⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ c = 3 - a - b b = 7 - 3 a 2 a = 4 (I) (II) (III)

$(III):$ $LHS / 2 = RHS / 2$

⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ c = 3 - a - b b = 7 - 3 a a = 2 (I) (II) (III)

$(II):$ $a = 2$

⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ c = 3 - a - b b = 7 - 3 (2) a = 2 (I) (II) (III)

$(II):$ Multiply

⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ c = 3 - a - b b = 7 - 6 a = 2 (I) (II) (III)

$(II):$ Subtract terms

⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ c = 3 - a - b b = 1 a = 2 (I) (II) (III)

$(I):$ $a = 2$ , $b = 1$

⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ c = 3 - 2 - 1 b = 1 a = 2 (I) (II) (III)

$(I):$ Subtract terms

⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ c = 0 b = 1 a = 2 (I) (II) (III)

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