1. Formulae and expressions
A4.1 Core Plenary
The rule for generating the next term in this sequence is
‘add the two previous terms’. …
Amy finds that when she takes the difference between terms for the first sequence she gets the same sequence back after a few terms!
1 (= 2 – 1) 1 (= 3 – 2) 2 (= 5 – 3) 3 (= 8 – 5) 5 (= 13 – 8) …
Does this happen with all sequences with the term-to-term rule ‘add the two previous terms’, whatever the first two terms are?
Use algebra to help you.
Preamble
This activity allow pupils to discover the fact that the term-by-term differences of a Fibonacci sequence form another Fibonacci sequence. The proof of this should be accessible (with some support) to most pupils, and may come as a surprise to many. After a few pupils have discovered the rule, a whole-class discussion would be useful. (The case where the second term is smaller than the first may need addressing.)
Possible content
Adding and subtracting simple algebraic expressions; simplifying expressions.
Resources
None.
Solution/Notes
Let the first terms be a and b. The sequence produced, using the rule is:
a b a + b a + 2b 2a + 3b 3a + 5b
Taking the difference between terms (by subtracting the first of the two terms each time) gives:
b – a a b a + b a + 2b …
The sequence regenerates itself.
Original Material © Cambridge University Press 2009
Original Material © Cambridge University Press 2009