The discussion revolves around finding the term $a_{2017}$ in a sequence defined by the initial condition $a_1 = 1$ and the recursive relationship given by the equation $$\frac{a_n}{n+1}=\frac{\sum_{i=1}^{n-1}a_i}{n-1}.$$ The scope includes mathematical reasoning and problem-solving related to sequences.
The discussion does not show clear consensus, as multiple participants have offered their solutions without resolving which is correct or if they align.
Details regarding the derivation steps and assumptions underlying the recursive relationship remain unspecified, leading to potential ambiguity in the solutions proposed.
Individuals interested in sequence problems, mathematical reasoning, and recursive relationships may find this discussion relevant.
lfdahl said:Find $a_{2017}$, if $a_1 = 1$, and $$\frac{a_n}{n+1}=\frac{\sum_{i=1}^{n-1}a_i}{n-1}---(1).$$
Albert said:$a_{2017}=1009\times 2^{2016}$ correct ?
Albert said:my solution:
from(1) we have:
$a_1=1,a_2=3,a_3=8,a_4=20,a_5=48,------$
so we may set :$a_n=2a_{n-1}+2^{n-2}---(2)$
or $a_n-a_{n-1}=a_{n-1}+2^{n-2}--(3)$
and $S_{n-1}=a_n(\dfrac{n-1}{n+1})---(4)$
so $$a_{2017}-{\sum_{i=1}^{2016}a_i}=2^{2016}$$
or $a_{2017}-S_{2016}=2^{2016}$
from $(3)(4)$$a_{2017}=1009\times 2^{2016}$