Good luck!

In summary, to prove that the sequence an+2=an+1+an is monotonically increasing, you need to show that ak+1 ≥ ak implies ak+2 ≥ ak+1 for all k ≥ 1.
  • #1
analysis001
21
0

Homework Statement


Prove that an+2=an+1+an where a1=1 and a2=1 is monotonically increasing.


Homework Equations


A sequence is monotonically increasing if an+1≥an for all n[itex]\in[/itex]N.


The Attempt at a Solution


Base cases:
a1≤a2 because 1=1.
a2≤a3 because 1<2.

Am I supposed to prove that an≤an+1 now? I'm not sure how to do that.
 
Last edited:
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  • #2
analysis001 said:

Homework Statement


Prove that an+2=an+1+an where a1=1 and a2=2 is monotonically increasing.


Homework Equations


A sequence is monotonically increasing if an+1≥an for all n[itex]\in[/itex]N.

The Attempt at a Solution


Base cases:
a1≤a2 because 1=1.
a2≤a3 because 1<2.

Am I supposed to prove that an≤an+1 now? I'm not sure how to do that.
a2 ≥ a1 because 2 ≥ 1 . After all, a2 = 2 and a1 = 1 .

Now, what you need to do:
Assume that the statement is true for some k, where k ≥ 1 .
I.e.:
Assume that ak+1 ≥ ak .​
From this, show that it follows that the statement is true for k+1.
I.e.:
Show that ak+2 ≥ ak+1 .​
 

1. What is the purpose of using proof by induction when dealing with sequences?

The purpose of using proof by induction is to prove that a statement or property holds for all elements in a sequence. It allows us to make a generalization about the entire sequence based on a few specific cases.

2. How does proof by induction work for sequences?

Proof by induction for sequences involves two steps:
1. Proving that the statement holds for the first element in the sequence
2. Assuming that the statement holds for an arbitrary element, and using this assumption to prove that it also holds for the next element in the sequence. This establishes the pattern and proves that the statement holds for all elements in the sequence.

3. Can proof by induction be used for all types of sequences?

Yes, proof by induction can be used for all types of sequences, including arithmetic, geometric, and recursive sequences. As long as the sequence has a defined pattern and can be broken down into individual elements, proof by induction can be applied.

4. What are the common mistakes to avoid when using proof by induction for sequences?

Some common mistakes to avoid when using proof by induction for sequences include:
- Assuming that the statement holds for all elements without first proving it for the first element
- Using circular reasoning, where the conclusion of the proof is used to prove the initial assumption
- Skipping steps or not clearly explaining the logic behind each step
It is important to carefully follow the two steps of proof by induction and clearly explain each step in order to avoid these mistakes.

5. Are there any real-world applications of proof by induction for sequences?

Yes, proof by induction for sequences has many real-world applications, especially in computer science and mathematics. It can be used to prove the correctness of algorithms and programs, analyze the time complexity of algorithms, and prove mathematical theorems and formulas. It is also commonly used in discrete mathematics, combinatorics, and graph theory.

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