<s>[INST] let $a_{n}=n(a_{1}+a_{2}+\dots+a_{n-1})$, $n\geqslant 2$, $a_{1}=1$. Find $a_{n}$ let $a_{n}=n(a_{1}+a_{2}+\dots+a_{n-1})$, $n\geqslant 2$, $a_{1}=1$. Find $a_{n}$
My Approach:
so I thought, since this is a homogeneous relation, I expanded the expression and wrote the characteristic equation of the same, since that was $n(x^1+x^2+\dots+x^{n-1})$. So I substituted the values of $n=2$, and my answer was $n(n-1)$ which is wrong.
EDIT: I cannot use the characteristic equation since the coeff is not constant
 [/INST] We have $$\frac{a_n}{n} = a_1+a_2+\cdots +a_{n-1}= (a_1+a_2+\cdots +a_{n-2}) + a_{n-1} = \frac{a_{n-1}}{n-1}+a_{n-1} = n\cdot \frac{a_{n-1}}{n-1}$$
which is valid for $n\ge 3$ (notice that it's not valid for $n=2$ because we used $a_{n-2}$).
Then $$\frac{a_n}{n}= n\cdot \frac{a_{n-1}}{n-1}=n(n-1)\cdot \frac{a_{n-2}}{n-2}=\cdots = n(n-1)\cdots 3\cdot \frac{a_{2}}{2} = \frac{n!}{2}$$
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