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An AIME competition question is:

Given $a,b,x,y \in \mathbb{R}$ and

$$a x + b y = 3, \quad a x^3 + b y^3 = 16$$

$$a x^2 + b y^2 = 7, \quad a x^4 + b y^4 =42$$

find

$$a x^5 + b y^5$$

The direct approach doesn't seem to work:

Solve[a x + b y == 3 \[And] a x^3 + b y^3 == 16 \[And] 
  a x^2 + b y^2 == 7 \[And] a x^4 + b y^4 == 42 \[And] 
  a x^5 + b y^5 == k, 
 {k}, Reals]

(Incidentally, the answer is $20$.)

What tricks or constraints or substitutions would allow Mathematica to solve this problem?

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  • $\begingroup$ One quick way: In[227]:= GroebnerBasis[{a*x + b*y - 3, a*x^3 + b*y^3 - 16, a*x^2 + b*y^2 - 7, a*x^4 + b*y^4 - 42, a*x^5 + b*y^5 - val}, val, {a, b, x, y}][[1]] Out[227]= -20 + val $\endgroup$ Aug 6 at 15:09
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Listing all your variables works:

k /. Solve[
a x + b y == 3 \[And] a x^3 + b y^3 == 16 \[And] 
 a x^2 + b y^2 == 7 \[And] a x^4 + b y^4 == 42 \[And] 
 a x^5 + b y^5 == k, {a, b, x, y, k}, Reals][[1]] // Simplify

Also, not listing any variables causes it to treat all free symbols as variables:

k /. Solve[a x + b y == 3 \[And] a x^3 + b y^3 == 16 \[And] 
a x^2 + b y^2 == 7 \[And] a x^4 + b y^4 == 42 \[And] 
a x^5 + b y^5 == k, Reals] // Simplify
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  • $\begingroup$ Oh... thanks! ($\checkmark$) $\endgroup$ Aug 5 at 17:19
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I think similar questions were asked and answered a lot.

Simplify[a*x^5 + b*y^5,Assumptions -> a*x + b*y == 3 && a*x^3 + b*y^3 == 16 && 
a*x^2 + b*y^2 == 7 &&   a*x^4 + b*y^4 == 42]

$20$

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  • separate the variable and parametric.
Solve[a x + b y == 3 ∧ a x^3 + b y^3 == 16 ∧ 
  a x^2 + b y^2 == 7 ∧ a x^4 + b y^4 == 42 ∧ 
  a x^5 + b y^5 == k, {k}, {a, b, x, y}]
  • useReduce
Reduce[a x + b y == 3 ∧ a x^3 + b y^3 == 16 ∧ 
  a x^2 + b y^2 == 7 ∧ a x^4 + b y^4 == 42 ∧ 
  a x^5 + b y^5 == k, {k}]
  • use ForAll
ForAll[{a, b, x, y}, 
 a x + b y == 3 ∧ a x^3 + b y^3 == 16 ∧ 
  a x^2 + b y^2 == 7 ∧ a x^4 + b y^4 == 42, a x^5 + b y^5 == k]
Resolve[%] // Simplify

Appendix

we can calculate all the r[n]=a x^n + b y^n if we provide the initial condition r[1] == 3, r[2] == 7, r[3] == 16, r[4] == 42.

By the uniqueness of such r[n],we only need to find a sequence that satisfied such condition. Here we use the second order difference equation.

rsol = RSolve[{r[n + 2] == p*r[n + 1] + q*r[n]}, r, n][[1]];
sol = Solve[{r[1] == 3, r[2] == 7, r[3] == 16, r[4] == 42} /. 
     rsol][[1]];
r /. rsol /. sol
% /@ Range[8] // Simplify

$$ \frac{2^{-n-1} \left(-14-2 \sqrt{87}\right)^n}{4263+457 \sqrt{87}}+\frac{1}{19} \left(49-\frac{19}{4263+457 \sqrt{87}}\right) 2^{-n-1} \left(2 \sqrt{87}-14\right)^n$$

{3, 7, 16, 42, 20, 1316, -17664, 297304}

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  • $\begingroup$ I like the completeness. Thanks. ($+1$) $\endgroup$ Aug 5 at 22:53
  • $\begingroup$ FindSequenceFunction[{3, 7, 16, 42, 20, 1316}] /@ Range[8] // Simplify $\endgroup$
    – cvgmt
    Aug 5 at 23:34
  • $\begingroup$ Or shorter Reduce[a x + b y == 3 \[And] a x^3 + b y^3 == 16 \[And] a x^2 + b y^2 == 7 \[And] a x^4 + b y^4 == 42 \[And] a x^5 + b y^5 == k]. $\endgroup$
    – user64494
    Aug 6 at 5:15

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