1975 Canadian MO Problems/Problem 2
Problem 2
A sequence of numbers satisfies
(i) ![$a_1 = \frac{1}{2}$](//latex.artofproblemsolving.com/7/c/9/7c93752fc9b032db99ff38f9114dcc254b098bcf.png)
![$a_1 = \frac{1}{2}$](http://latex.artofproblemsolving.com/7/c/9/7c93752fc9b032db99ff38f9114dcc254b098bcf.png)
(ii) ![$a_1+a_2+\cdots+a_n=n^2a_n\quad(n\ge1).$](//latex.artofproblemsolving.com/a/f/2/af2aa146d3f29ed75142daa3c1a4819df46b0582.png)
![$a_1+a_2+\cdots+a_n=n^2a_n\quad(n\ge1).$](http://latex.artofproblemsolving.com/a/f/2/af2aa146d3f29ed75142daa3c1a4819df46b0582.png)
Determine the value of
Solution
I claim with a proof by induction. First we can use partial fraction decomposition to rewrite
as
. We have
We can set coefficients equal,
and
. Now,
Base Case: If , then
So,
when
.
Inductive Step: Suppose conclusion is true for , such that we have
We also have
Add
to both sides. The left side becomes
which is a telescoping series equal to
. Now, we have
We have
thus the conclusion being true for
, implies that it holds for
, so our induction is complete.
1975 Canadian MO (Problems) | ||
Preceded by Problem 1 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • | Followed by Problem 3 |