48
High school
Again & Again
• • • • • • • • • • • Using the activity in the classroom • • • • • • • • • • •
This activity could be introduced orally. Ask students to think of a number and allow them to
generate the sequence using this number as the first term.
Students can investigate different starting values, discovering that all sequences will converge to 2.
More able students could be asked to prove this, i.e.:
x
÷
2 +
1
=
x
x
+ 2 = 2
x
2 = 2
x
-
x
2 =
x
Students can investigate different rules such as
÷
3 +
1
, or
÷
2 + 5, generalizing which rules con-
verge to an integer limit.
Generally, the sequence
÷
a
+
b
, converges to
ab
/(
a
-
1
), where
a
≠
1
and
a
≠
0.
• • • • • • • • • • • • • • • Points for students to discuss • • • • • • • • • • • • • •
It may be useful to show students how to generate sequences on the calculator.
Further Ideas
• Students use their calculators to generate different convergent sequences. They then have
to try and identify the rules that each other used.
• Find rules to define sequences that converge to a given limit. For example, find different
sequences that converge to 3.
Summary of Contents for EL-531RH
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