Question

The sequence a(n) satisfies a(1) = 3 and

a(n) = [a(n-1)-1]/a(n-1)+1

for all n ≥ 2. Find a(100).

Answer 1

To find a(100), we first need to find a(2), a(3), a(4), and so on until we reach a(100) by applying the recursive formula given.

a(1) = 3

To find a(2):
a(2) = [a(1)-1]/a(1)+1
a(2) = [3-1]/3+1
a(2) = 2/4
a(2) = 1/2

To find a(3):
a(3) = [a(2)-1]/a(2)+1
a(3) = [1/2-1]/1/2+1
a(3) = -1/2 / 3/2
a(3) = -1/3

Continuing on in this fashion, we find:

a(4) = 1/2
a(5) = 1
a(6) = 0

It's clear that the sequence oscillates between a positive number and zero.

Therefore, for n ≥ 5, a(n) = 1 for odd n and a(n) = 0 for even n.

Since 100 is even, a(100) = 0.