Let a sequence {a_n } satisfies { (( a_1 =1)),((na_n =n+2Σ_(k=1) ^(n−1) a_k , n>2)) :} Find the value of a


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Question Number 129413 by ZiYangLee last updated on 15/Jan/21
$Let a sequence {a_n } satisfies { (( a_1 =1)),((na_n =n+2Σ_(k=1) ^(n−1) a_k , n>2)) :} Find the value of a_(2021) .$
$Let a sequence {a_{n}} satisfies$ ${\begin{cases} a_{1} = 1 \\ {n a}_{n} = n + 2 \underset{k = 1}{\sum^{n - 1}} a_{k}, n > 2 \end{cases}$ $Find the value of a_{2021} .$
Commented byJDamian last updated on 15/Jan/21

$a_{2} i s n o t d e f i n e d, {i s n}^{'} t i t ?$
	Answered by Olaf last updated on 15/Jan/21

	$2 a_{2} = 2 + 2 a_{1} = 4 \Rightarrow a_{2} = 2$ $3 a_{3} = 3 + 2 (a_{2} + a_{1}) = 9 \Rightarrow a_{3} = 3$ $4 a_{4} = 4 + 2 (a_{3} + a_{2} + a_{1}) = 16 \Rightarrow a_{4} = 4$ $5 a_{5} = 5 + 2 (a_{4} + a_{3} + a_{2} + a_{1}) = 25 \Rightarrow a_{5} = 5$ $a_{n} = n ?$ ${It}^{'} s true for n = 1 : a_{1} = 1$ $Suppose a_{k} = k, k ⩽ n - 1, then by induction :$ ${n a}_{n} = n + 2 \underset{k = 1}{\sum^{n - 1}} a_{k} = n + 2 \underset{k = 1}{\sum^{n - 1}} k$ ${n a}_{n} = n + 2 \frac{(n - 1) n}{2} = n^{2}$ $\Rightarrow a_{n} = n$ $a_{2021} = 2021$
	Commented byJDamian last updated on 16/Jan/21

	$b u t t h a t m e t h o d i s f o r n > 2, n o t f o r n = 2$
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