<s>[INST] If $ax+by=7$, . 
If 
  $$ax+by=7$$
  $$ax^2+by^2=49$$
  $$ax^3+by^3=133$$
  $$ax^4+by^4=406$$
  then find the value of
  $$2014(x+y-xy) - 100(a+b)$$

My attempt:
$$ax^2+by^2=49$$
$$ax^2+by^2=(ax+by)^2$$
$$ax^2+by^2=a^2x^2+2abxy+b^2y^2$$
$$ax^2-a^2x^2+by^2-b^2y^2=2abxy$$
$$ax^2(1-a)+by^2(1-b)=2abxy$$
 [/INST] Let $\frac{ax}{ax+by}=\alpha$ and $\frac{by}{ax+by}=\beta$.
Thus, $$\alpha+\beta=1,$$
$$\alpha x+\beta y=7,$$
$$\alpha x^2+\beta y^2=19$$ and
$$\alpha x^3+\beta y^3=58.$$
Hence, $$(x+y)(\alpha x+\beta y)=7(x+y)$$ or
$$19+xy(\alpha+\beta)=7(x+y)$$ or
$$19+xy=7(x+y).$$
In another hand
$$(x+y)(\alpha x^2+\beta y^2)=19(x+y)$$ or
$$58+xy(\alpha x+\beta y)=19(x+y)$$ or
$$58+7xy=19(x+y).$$
From here we obtain $x+y=2.5$ and $xy=-\frac{3}{2}$ and the rest is smooth.
I got $a+b=21$ and $$2014(x+y-xy)-100(a+b)=2014\cdot4-100\cdot21=5956.$$ 
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