Expert: Josh - 4/22/2004

Question
How do i find the next term when i can not find a pattern?


Such as 1/3, 1/4, 9/7, 16/9

Answer
This is neither an arithmetic nor geometric sequence, because it is neither monotonic increasing nor monotonic decreasing.

The difference between two consecutive terms does not yield a constant amount (the requirement for an arithmetic series, A.S) and they are not related by a constant factor (the requirement for a geometric series, G.S)

This is what I think in this case,
let T(i)=n(i)/d(i) represent the ith term,
   n(i) and d(i) represent the numerator and denominator respectively; and n=1,2,3,4,...

So, T(1)=1/3, n(1)=1, d(1)=3.
   T(2)=1/4, n(2)=1, d(2)=4.
Observe that, n(i)=d(i-2)*d(i-2) ...[#1]
and           d(i)=2*d(i-2)+1    ...[#2]
for index i > 2.

To verify this, start with i=3,
according to the relation [#1],
n(3)=d(1)*d(1)=3*3=9
d(3)=2*d(1)+1 =2*3+1=7
which is in agreement with the third term, T(3).

Similarly,
n(4)=d(2)*d(2)=4*4=16
d(4)=2*d(2)+1 =2*4+1=9
agree with T(4).

It's not obvious, is it?

Cheers:)