A sequence u0, u1, u2,
... is such that un+1 = 0.9un +
90.
Show that
un = 0.9n(u0 - 900) +
900.
A recurrence relation is given, and we are asked to deduce an
explicit (closed form) formula for un.
For this simple kind of
example, I would look directly for a pattern:
u0 = u0
u1 = 0.9(u0) + 90
u2 = 0.92(u0) + 0.9(90) +
90
u3 = 0.93(u0) + 0.92(90)
+ 0.9(90) + 90
...
un =
0.9n(u0) +
0.9n-1(90) + 0.9n-2(90) + ...
+ 0.9(90) + 90
= 0.9n(u0) +
0.9n-1(90)
+ 0.9n-2(90) + ...
+ 0.9(90) +
90
Therefore:
un = 0.9n(u0) + Sum
of a certain GP
= Correct answer
For this particular example, is there a better way to find
un, other than by spotting the pattern? For teaching
purposes, I wonder if there is a common and elegant technique that
I am not aware of. I
don't mean math induction, method of differences, or anything like
that, which I don't think is needed for a simple example like this
one. I mean a straightforward algebraic way to get a formula
for un from the given recurrence
relation.
