Question

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

Answer 1

I think you mean $a_n= \frac{a_{n-1}}{a_{n-2}}$
basically that measn yo divide that term by the previous term
ok

a3=a1/a2=1/2
a4=a3/a2=(1/2)/2=1/4
a5=a4/a3=(1/4)/(1/2)=1/2
a6=a5/a4=(1/2)/(1/4)=2
a7=a6/a5=2/(1/2)=4
a8=a7/a6=4/2=2
a9=a8/a7=2/4=1/2
a10=a9/a8=(1/2)/2=1/4
ok, so notice a pattern
a3=a9
a4=a10
I therefor conclude that $a_n=a_{n+6w}$ where w is a whole number
and n>2ok, so
241=6*39+7
so therefor
$a_{241}=a_{7+6(39)}$
therefor the 241th term is same as a7 or 4

I say a241 is 4

Answer 2

Answer:

$a_{241} = 1$

Step-by-step explanation:

$a_{1}=1\\a_{2}=2\\\\a_{n}=\frac{a_{n-1}}{a_{n-2}}\\\\a_{3}=\frac{a_2}{a_1}=\frac{1}{2}\\\\a_{4}=\frac{a_3}{a_2}=\frac{\frac{1}{2}}{2}=\frac{1}{4}\\\\a_{5}=\frac{a_4}{a_3}=\frac{\frac{1}{4}}{\frac{1}{2}}=\frac{1}{2}\\\\a_{6}=\frac{a_5}{a_4}=\frac{\frac{1}{2}}{\frac{1}{4}}=2}\\\\a_{7}=\frac{a_6}{a_5}=\frac{\frac{1}{2}}{2}=4\\\\a_{8}=\frac{a_7}{a_6}=\frac{4}{2}=2\\\\a_{9}=\frac{a_8}{a_7}=\frac{2}{4}=\frac{1}{2}\\\\a_{10}=\frac{a_9}{a_8}=\frac{\frac{1}{2}}{2}=\frac{1}{4}$

Now, as observed the pattern is getting repeated

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

$So,a_{241}=1$