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}$
The sequence challenge is a mathematical problem where you are given a sequence of numbers and are asked to find a specific term within that sequence, in this case, the term at position 2017.
To find the term at position 2017, you need to first examine the pattern of the sequence and try to find a rule or formula that describes how the numbers are changing. Once you have a formula, you can plug in 2017 as the input to find the corresponding output, which will be the desired term.
While some mathematical background may be helpful, anyone with basic knowledge of arithmetic and algebra can attempt to solve the sequence challenge. It mainly requires critical thinking and problem-solving skills.
There are various strategies and techniques that can be used to solve the sequence challenge, such as finding the differences between consecutive terms, looking for common factors or multiples, or using geometric or recursive patterns. It may also be helpful to look at smaller sections of the sequence to identify any repeating patterns.
If you are unable to find a pattern or formula for the sequence, you can try brute force or trial and error methods by plugging in different values for the position until you find the desired term. Alternatively, you can also ask for help from others or use online resources to see if someone has already solved the challenge.