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AMC 10 USA Math Olympiad

Problem on Series and Sequences | SMO, 2012 | Problem 23

Try this beautiful problem from Singapore Mathematics Olympiad, 2012 based on Series and Sequences. You may use sequential hints to solve the problem.

Try this beautiful problem from Singapore Mathematics Olympiad, 2012 based on Series and Sequences.

Problem on Series and Sequences (SMO Test)


For each positive integer \(n \geq 1\) , we define the recursive relation given by \(a_{n+1} = \frac {1}{1+a_{n}} \).

Suppose that \(a_{1} = a_{2012}\).Find the sum of the squares of all

possible values of \(a_{1}\).

  • 2
  • 3
  • 6
  • 12

Key Concepts


Series and Sequence

Functional Equation

Recursive Relation

Check the Answer


Answer: 3

Singapore Mathematics Olympiad

Challenges and Thrills – Pre – College Mathematics

Try with Hints


If you got stuck start from here :

At first we have to understand the sequence it is following

Given that : \(a_{n+1} = \frac {1}{1+a_{n}} \)

Let \(a_{1} = a\)

so , \(a_{2} = \frac {1}{1 + a_{1}}\) = \(\frac {1}{1 + a}\)

Again, \(a_{3} = \frac {1}{1+a_{2}} = \frac {1+a}{2+a} \)

For , \(a_{4} = \frac {1}{1+a_{3}} = \frac {2+a}{3+2a} \)

And , \(a_{5} = \frac {1}{1+a_{4}} = \frac {3+2a}{5+3a} \)

and so on……..

Try to do the rest …………………………….

Looking at the previous hint ………………

In general we can say ……………..

\(a_{n} = \frac {F_{n} + F_{n-1}a}{F_{n+1} +F_{n}a}\)

Where \(F_{1} = 0 , F_{ 2} = 1\) and \(F_{n+1} = F_{n} \) for all value of \(n\geq 1\)

Try to do the rest …….

Here is the rest of the solution,

If \(a_{2012} = \frac {F_{2012}+F_{2011}a}{F_{2013} + F{2012}a} = a \)

Then \((a^2+a-1 )F_{2012} = 0\)

Since \(F_{2012}>0\) we have \(a^2 +a -1 = 0\) ……………………….(1)

Assume x and y are the two roots of the \(eq^n (1)\), then

\(x^2 + y^2 = (x+y)^2 -2xy = (-1)^2 – 2(-1) = 3\) (Answer)

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