2019 AMC 12A Problem 17

Problem 17 of 25IntermediateAlgebra

Let

s_k

denote the sum of the

k

th powers of the roots of the polynomial

x^3 - 5x^2 + 8x - 13.

In particular,

s_0 = 3,

s_1 = 5,

and

s_2 = 9.

Let

a,

b,

and

c

be real numbers such that

s_{k+1} = a\,s_k + b\,s_{k-1} + c\,s_{k-2}

for

k = 2,

3,

\ldots.

What is

a + b + c?

Show solution

Solution

Every root

r

satisfies

r^3 = 5r^2 - 8r + 13,

so

r^{k+1} = 5r^k - 8r^{k-1} + 13r^{k-2}.

Summing over the three roots gives

s_{k+1} = 5s_k - 8s_{k-1} + 13s_{k-2},

so

a = 5,

b = -8,

c = 13.

Therefore

a + b + c = 5 - 8 + 13 = 10.

Thus, the correct answer is D.

Tagged: Newton’s Sums · polynomial · recursion

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