Find ax^5+by^5 if the real numbers a, b, x, and y satisfy the equations ax+by&=3, ax^2+by^2&=7, ax^3+by^3&=16, ax^4+by^4&=42.

1990 AIME Problem 15

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15.

Find

ax^5+by^5

if the real numbers

a,

b,

x,

and

y

satisfy the equations

\begin{aligned}ax+by&=3,\\ax^2+by^2&=7,\\ax^3+by^3&=16,\\ax^4+by^4&=42.\end{aligned}

Answer

Let

S_k=ax^k+by^k

and derive a recurrence using

x+y

and

xy

Use

S_3

and

S_4

to solve for the two recurrence coefficients before computing

S_5

Let

S_k=ax^k+by^k,

p=x+y,

and

q=xy.

Since

x

and

y

each satisfy

t^2=pt-q,

S_{k+2}=pS_{k+1}-qS_k.

Using

S_1=3,

S_2=7,

S_3=16,

and

S_4=42

gives

\begin{aligned}7p-3q&=16,\\16p-7q&=42.\end{aligned}

Solving yields

p=-14

and

q=-38.

Therefore

\begin{aligned}S_5&=pS_4-qS_3\\&=-14(42)+38(16)\\&=20.\end{aligned}