The sequence {a
_{n}} is defined by
a
_{1} = 1 and a
_{n+1} = a
_{n}+1/(a
_{n}^{2}).
Show that a_{2016} is over 18.
We are going to consider how many values it takes for the series to increase by 1, say from 2.xxx to 3.xxx, using the closest result to the integer.
The following table shows the integer reached, the entry at which it was reached, and the difference between the entries. The last column should be pretty familiar; in fact it is just n(n+1). The second column therefore has the equation (2*n  3*n^2 + n^3)/3, which is worth 1938 when n=18 and 2280 when n=19.
2 2 2
3 8 6
4 20 12
5 40 20
6 70 30
7 112 42
8 168 56
9 240 72
etc.
The equation is not exact as values increase, but the discrepancy is less than 1 part in 200. Since 2016 is far more than 1938 and far less than 2280, a2016 is larger than 18 and less than 19.

Posted by broll
on 20170330 10:21:30 