1 Background Info A continued fraction is a structure created by using fractions of fractions in an iter- ative manner. An iterative process is one where there is an obvious pattern that lets you find the next term. For example the sum 1 + 1+ 1 + ... intuitively tells us that we are simply adding 1 every time. Or the sum 1+ + + +... tells us that we are adding the reciprocals of natural numbers. When a similar process is applied to the definition of a fraction, we obtain a continued fraction. In particular, we will be examining the following sequence of fractions: 1,1 + 7. 1+ 1 + + 2 Exploring values of continued fraction and application of quadratic formula 1. Using the rules you learnt about fractions of fraction, evaluate the above given 5 expressions as a simplified fraction. (A simplified fraction is of the form .) 2. Look at the numerator and denominator of each of the fractions obtained above - what pattern do you notice ? Based on the pattern, can you predict what the next fraction should be without actually calculating it? 3. Once you have the pattern, can you find the next 5 fractions. This means that in total you should have 10 fractions now - 5 from the calculation and 5 more from the pattern observed. 4. Find the decimal values of each of these fractions and make a comment on the value that you see. 5. Justify this statement: If this process goes on forever, then the expression can be represented by the equation = = 1 +6. Using methods learnt in class, solve the above equation for a - since it is a quadratic, it will give two solutions but only one makes sense. Which one makes sense and why ? 7. Compare this value to the observation you made in (4) 8. Both the numbers found in (6) appear in Worksheet 9 as ....... of each other