18.3.1. Unique Factorization

Try these first

1. 1.

   Define a subsequence.

   Hint

   Think about examples of both sequences and subsequences.

2. 2.

   Is the sequence [4, 3, 2, 1] strictly increasing or decreasing?

   Hint

   Check the relationship between each consecutive term.

3. 3.

   What is a strictly increasing sequence?

   • A sequence where each term is less than the following term
   • A sequence where each term is greater than the previous term
   • A sequence sorted in descending order

   Hint

   Look out for terms relative to one another.

4. 4.

   True or False: Every sequence of n + 1 distinct real numbers contains a decreasing subsequence of length n + 1.

   • True
   • False

   Hint

   Consider subsequences formed from increasing distinct numbers.

5. 5.

   Given the sequence [9, 7, 5, 8, 10], demonstrate that there exists either a strictly increasing or decreasing subsequence.

   Hint

   Look at the sequence order closely, not all need to be taken together.

6. 6.

   Using a custom sequence of 7 distinct numbers, explain how you can derive an increasing subsequence of at least 5.

   Hint

   Choose wisely to increase; several must be smaller than your larger picks.