# Lower Bound

## Problem

## Getting Started

Can you spot a pattern in the answers?

All the answers are 2 and a bit and that bit seems to get smaller. What can you say about it? Can you explain why the pattern arises?

## Student Solutions

**Vassil Vassilev** , a 14 year old Bulgarian
student from St Michael's College, Leeds, sent the following
solution:

Element No 1 $$ {1\over 2} + {2\over 1} = {1 + 4\over 2} = {5\over 2} = 2 + {1\over 2} = 2 + {1\over 1\times(1+1)} $$ Element No 2 $$ {2\over 3} + {3\over 2} = {4 + 9\over 6} = {13\over 6} = 2 + {1\over 6} = 2 + {1\over 2\times(2+1)} $$ Element No 3 $$ {3\over 4} + {4\over 3} = {9 + 16\over 12} = {25\over 12} = 2 + {1\over 12} = 2 + {1\over 3\times(3 + 1)} $$ Element No 4 $$ {4\over 5} + {5\over 4} = {16 + 15\over 20} = {41\over 20} = 2 + {1\over 20} = 2 + {1\over 4\times(4 + 1)} $$ Element No n

Some, but not all, of the points on the graph of $$ y = x + {1\over x} $$ represent terms of the sequence. You might like to draw the graph and look at what happens to it around $x = 1$.

Vassil worked out the $n$th term of the similar sequence formed by adding the squares of these fractions and after doing some algebra to simplify the expression he obtained the following result: $$ \left({n\over n + 1}\right)^2 + \left({n + 1\over n}\right)^2 = 2 + \left({2n + 1\over n(n + 1)}\right)^2. $$ Here again the limit as n tends to infinity is 2. You might like to prove this result in another way.

## Teachers' Resources

Younger learners can work with numbers and find out what happens to the sequence.

As you work out more terms does the value seem to get closer to a particular number? Are the values always greater than that number? Why do you think this happens?

If you want to go on to use algebra, a bit of algebra will help you to explain what happens and to prove that there is a limiting value for this sequence. However you will have to know how to multiply two bracketed terms together to do this so it might be something to come back to when you have learnt a little more algebra.