Question

A sequence {an} is defined recursively, with a1 = 1, a2 = 2 and, for n > 2, an =

an-1
an-2
. Find the term a241.

Answer 1

`a_(241) = 1`

Explanation:

`a_(1)=1\\a_(2)=2\\\\a_(n)=(a_(n-1))/(a_(n-2))\\\\a_(3)=(a_2)/(a_1)=(1)/(2)\\\\a_(4)=(a_3)/(a_2)=((1)/(2))/(2)=(1)/(4)\\\\a_(5)=(a_4)/(a_3)=((1)/(4))/((1)/(2))=(1)/(2)\\\\a_(6)=(a_5)/(a_4)=((1)/(2))/((1)/(4))=2}\\\\a_(7)=(a_6)/(a_5)=((1)/(2))/(2)=4\\\\a_(8)=(a_7)/(a_6)=(4)/(2)=2\\\\a_(9)=(a_8)/(a_7)=(2)/(4)=(1)/(2)\\\\a_(10)=(a_9)/(a_8)=((1)/(2))/(2)=(1)/(4)`

Now, as observed the pattern is getting repeated

`a_3 = a_9\\a_4=a_(10)\\\therefore a_n=a_(n+6\cdot w)\thinspace ;\text{w is a whole number}\\\implies a_(241)=a_(1+6\cdot 40)\\ \implies a_(241)=a_1=1`

`So,a_(241)=1`