# Proof by Induction- Sequences

## 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$\in$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.

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## 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$\in$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 .​