Difference between revisions of "2018 UNCO Math Contest II Problems/Problem 9"
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(b) Find a recursion formula for <math>G(n + 1)</math>. That is, find a formula that expresses <math>G(n + 1)</math> in | (b) Find a recursion formula for <math>G(n + 1)</math>. That is, find a formula that expresses <math>G(n + 1)</math> in | ||
| − | terms of <math>G(n), G(n | + | terms of <math>G(n), G(n - 1),\dots</math> |
(c) Give an explanation that shows that the formula you give is correct. | (c) Give an explanation that shows that the formula you give is correct. | ||
| Line 15: | Line 15: | ||
a) <math>G(3) = 5, G(4) = 8, G(5) = 13</math> | a) <math>G(3) = 5, G(4) = 8, G(5) = 13</math> | ||
| − | b) <math>G(n) = G(n | + | b) <math>G(n) = G(n - 1) + G(n - 2); G(0) = 1 \text{ and } G(1) = 2</math> |
| − | c) <math>G(n + 1) = G(n) + G(n | + | c) <math>G(n + 1) = G(n) + G(n - 1)</math>. |
== See also == | == See also == | ||
Latest revision as of 20:02, 28 October 2023
Problem
Call a set of integers Grassilian if each of its elements is at least as large as the number of
elements in the set. For example, the three-element set 
is not Grassilian, but the
six-element set 
is Grassilian. Let 
be the number of Grassilian subsets of 
. (By definition, the empty set is a subset of every set and is Grassilian.)
(a) Find 
, 
, and 
.
(b) Find a recursion formula for 
. That is, find a formula that expresses in
terms of 
(c) Give an explanation that shows that the formula you give is correct.
Solution
a) 
b) 
c) 
.
See also
| 2018 UNCO Math Contest II (Problems • Answer Key • Resources) | ||
| Preceded by Problem 8 |
Followed by Problem 10 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 | ||
| All UNCO Math Contest Problems and Solutions | ||
