Investigating Fractions Chains

Students are asked to find the sums of convergent geometric series (although this term is not used in the questions). The aim is that, by investigating what happens to the sum as increasing numbers of terms are added, the students can discover the limit for the sum (again, this terminology is not used). I wrote it with able Year 7 pupils in mind to extend the topic of fractions and encourage investigatory approaches to learning but it would be appropriate for many KS3 or KS4 pupils (and could be used as an investigation to introduce convergent geometric series in KS5). Some questions involve the use of algebra for generalising and, while some students may have limited experience with this, it may serve as a useful introduction to the use of algebra to describe patterns. I have tried to push the need for clear record-keeping and observation and asked the students to comment on any interesting patterns or surprises.