A sequence {a_n} is defined recursively, with a_1 = 1, a_2=2 and, for n>2, a_n = (a_(n-1))/(a_(n-2)). How do you find the term a_(241)?

1 Answer
Jun 7, 2016

a_241 = 1

Explanation:

First let's observe some values of the sequence to find a common trend.

a_3 = (a_2)/(a_1) = 2/1 = 2
a_4 = (a_3)/(a_2) = 2/2 = 1
a_5 = (a_4)/(a_3) = 1/2
a_6 = (a_5)/(a_4) = (1/2)/1=1/2
a_7 = (a_6)/(a_5) = (1/2)/(1/2) = 1 = a_1
a_8 = (a_7)/(a_6) = 1/(1/2) = 2 = a_2

As you probably notice now, the sequence repeats every 6 terms, meaning that a_1 = a_7 = a_13 = ... a_(1+6t).

In fact, we can generalize. If 1<=k<7, then

a_k = a_(k+6) = a_(k+12) = ... = a_(k+6t).

What this means that for any n, we can find which value a_n corresponds to by taking the remainder when we divide n by 6.

In the case of n=241, 241/6 = (240+1)/6 = 40 1/6, which implies a remainder of 1 and thus a_241 corresponds to a_1, which is 1.