Linear Recurrence Relations

A Maths Problem

Today’s Challenge

(An) and (Bn) are two sequences defined recursively as follows.

with A0 = B0 = 0.

Find the following ratio

Give this problem a try before jumping in for the solution!

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Solution

The way I solved this problem is to start by playing around with the terms A2016 and B672. I expanded the terms respectively to find a pattern.

We will expand the term by considering 2A2015

We will iterate the process one more time to find a pattern

We can deduce that for the general An, we have

This means for A2016, we have

Now that we have found A2016, try and use a similar technique to find B627 before reading on!