Each student in my year 8 class was asked to consider this problem for homework and present their findings. I have submitted the best two entries but a summary of other interesting ideas are listed below.
Both Sam Huston and Robbie Waddell did not go with an exact version of any of the methods but instead looked for a case when 1000/difference gave an answer of something.5
These ideas are attached- Sam explains why and Robbie applied this idea for 1000.
It was also interesting to see the varying ways of writing a formula for the nth term of A_m
Andrea, Marjolaine and Misha all gave it as 2^m + (n-1)*2^(m+1)
Shunya gave it as (2+4(n-1))*2^(m-1)
Dmitris gave it as (2n-1)*2^m
Marcus gave it as (4n-2)*2^(n-1)
Alex gave it as n*2^(m+1)-2^m
I hope these ideas are of interest to you!
Thanks,
Andy Bottomley