• Support PF! Buy your school textbooks, materials and every day products Here!

Proof by Induction- Sequences

  • #1

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:

Answers and Replies

  • #2
SammyS
Staff Emeritus
Science Advisor
Homework Helper
Gold Member
11,312
1,001

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 .​
 

Related Threads on Proof by Induction- Sequences

  • Last Post
Replies
3
Views
1K
Replies
11
Views
19K
Replies
2
Views
2K
Replies
13
Views
2K
  • Last Post
Replies
16
Views
2K
  • Last Post
Replies
4
Views
781
  • Last Post
Replies
9
Views
1K
  • Last Post
Replies
4
Views
1K
  • Last Post
Replies
5
Views
1K
  • Last Post
Replies
3
Views
880
Top