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Sequences defined Recursively
STEP 1998, Math III, #3: The value
of a bond after days is determined by the equation
where
and are given constants. By looking for solutions of the form for some constants , , and ,
or otherwise, find
in terms of . [Hint: You're being asked to find , , and in terms of , , and so that the above two equations are both true.]
What is the solution for
? Show that this is the limit (for fixed ) as of your solution for .
[Hint: Think about what the symbols means and what is implied by the situation of
.]
Skill Set
Recursives Sequences
No. 1
STEP 2006, Math II, #1: The sequence of real numbers
, , , ... is defined by
, and for (*)
where
is a constant.
(i) Determine the values of
for which the sequence (*) is:
(a) constant;
(b) periodic with period 2;
(c) periodic with period 4.
(ii) In the case
, show that for all . Given that in this case the sequence (*) converges to a limit , find the value of .
Red Question: The first term in a sequence of numbers is
. Succeeding terms are defined by the statement for . Find the value of
.
STEP 2008, Math II, #1: A sequence of points , , ... in the cartesian plane is generated by first choosing then applying the rule, for
where
and are given real constants.
(i) In the case
and , find the values of for which the sequence is constant.
(ii) Given that , find the values of and for which the sequence has period 2.
STEP 2004, Math I, #8: A sequence
, , , ... is said to be strictly increasing if for all .
(i) The terms of the sequence
, , , ... satisfy
for
. Prove that if then the sequence is strictly increasing.
(ii) The terms of the sequence
, , , ... satisfy
for
. Prove that if then the sequence is strictly increasing but that for all .
[Hint: Be careful about the order of your statements. You absolutely MAY NOT assume the inequality is true and show there is no contradiction.]
STEP 2002, Math II, #5: The numbers
, where , satisfy the recursive definition .
(i) Prove that,
and , then for all .
(ii) Given instead that
, with and , find in terms of .
(iii) Given instead that
, with and , show that
,
where
. Given in addition that , find the possible values of in terms of .
STEP 1999, Math I, #3: The
positive numbers where , satisfy
, , . . . , ,
and also
.
Show that
(i) ,
(ii) ,
(iii)
Hence find the value of
.
STEP 2007, Math III, #3: (Fibonacci) A sequence of numbers, , , ... , is defined by , , and
for .
(i) Write down the values of , , ... , .
(ii) Prove that .
(iii) Prove by induction or otherwise that and deduce that is divisible by .
(iv) Prove that is divisible by .
STEP 2012, Math II, #8: The positive numbers
, and satisfy . Show that
.
The sequence
, , . . . is defined by , and
( )
where
, and are given positive numbers (and and are such that no term in the sequence is zero).
Prove that
for some number
which you should express in terms of , and .
Hence, or otherwise, show that if , the sequence is strictly increasing (that is, for all .)
Comment on the case
.
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