Question

Describe the rule for the sequence 2, 1, 2/3, 1/2, 2/5, 1/3, 1/7,...

Answer 1

Divide 2 by n.

Explanation:

Answer 2

Multiply 2 by 1/2 to get 1.

Multiply 1 by 2/3 to get 2/3.

Multiply 2/3 by 3/4 to get 6/12 = 1/2.

Multiply 1/2 by 4/5 to get 4/10 = 2/5.

Multiply 2/5 by 5/6 to get 10/30 = 1/3.

Multiply 1/3 by 6/7 to get 6/21 = 2/7. (I suspect there's a typo in the question.)

And so on, so that the nth term in the sequence is multiplied by n/(n + 1) to get the (n + 1)th term.

Recursively, the sequence is given by

`\begin{cases}a_1=2\\a_n=\frac{n-1}na_(n-1)&\text{for }n>1\end{cases}`

We can solve this exactly by iterating:

`a_n=\frac{n-1}na_(n-1)=\frac{n-1}n\frac{n-2}na_(n-1)=\frac{n-1}n(n-2)/(n-1)(n-3)/(n-2)a_(n-3)=\cdots`

and so on down to

`a_n=((n-1)\cdot(n-2)\cdot(n-3)\cdot\cdots\cdot3\cdot2\cdot1)/(n\cdot(n-1)\cdot(n-2)\cdot\cdots\cdot4\cdot3\cdot2)a_1`

or

`a_n=((n-1)!)/(n!)a_1`

and with lots of cancellation, we end up with

`a_n=\frac{a_1}n=\boxed{\frac2n}`