Integral calculus: sequences and recursively defined functions
Download file: RecursiveSequence.pg
DOCUMENT();
loadMacros('PGstandard.pl', 'PGML.pl', 'parserFunction.pl', 'PGcourse.pl');
Preamble
A new named function will be defined and added to the context. This can be done using parserFunction.pl.
Context()->variables->are(n => 'Real');
parserFunction(f => 'sin(pi^n) + e * n^2');
$fn = Formula('3 f(n - 1) + 2');
Setup
Define a new named function f as something the student
is unlikely to guess. The named function f is just a
placeholder since the student will enter expressions involving
f(n - 1). It will be interpreted internally as defined
here, and the only thing the student sees is f(n - 1).
If the recursion has a closed-form solution (e.g., the Fibonacci
numbers are given by f(n) = (a^n - (1 - a)^n) / sqrt(5)
where a = (1 + sqrt(5)) / 2) and you want to allow students
to enter the closed-form solution, it would be good to define
f using that explicit solution in case the student tries to
answer the question by entering the explicit solution.
BEGIN_PGML
If [`f(n)`] defines a sequence for all integegers [`n \geq 0`] that satisfies
the property that [`f(n)`] is two more than three times the previous value.
Find a recursive definition for [`f(n)`].
[`f(n) =`] [_]{$fn}{15}
END_PGML
Statement
This is the problem statement in PGML.BEGIN_PGML_SOLUTION Solution explanation goes here. END_PGML_SOLUTION ENDDOCUMENT();
Solution
A solution should be provided here.