<s>[INST] Aptitude simplification question Given $$p+q+r=1\;\;\;\;\&\;\;\;p^2+q^2+r^2=2\;\;\;\;\&\;\;\;p^3+q^3+r^3=3$$
find the value of  $$p^4+q^4+r^4$$
 [/INST] More generally, we have that for all $a,b,c\in\mathbb R$, $n\in\mathbb Z$, $n\ge 0$:
$$S_{n+3}=S_{n+2}S_1-S_{n+1}(ab+bc+ca)+S_{n}(abc),$$
where $S_n=a^n+b^n+c^n$. It can be proved by simply expanding.
Also notice $ab+bc+ca=\frac{(a+b+c)^2-\left(a^2+b^2+c^2\right)}{2}$.
Now apply this for your problem. Let $(a,b,c)=(p,q,r)$ and also let firstly $n=0$ (to find $pqr$) and then $n=1$ (to find $p^4+q^4+r^4$).
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