COSMO · STUDY + TEACHING DECK

Sequences

March 7, 2026

A question worth sharing

Recurrence Relations

Define a sequence by how each new term depends on earlier ones.

ThinkCompareExplain

Today’s route

A question becomes an explanation.

Try

Enter through a problem. A partial attempt counts.

Connect

Use recurrence, initial condition, closed form to name what you notice.

Convince

Build an argument a partner can check.

The goal is a method someone else can use.

The opener

Start with a question.

Opening problem

A sequence starts 2, 5 and each term is the sum of the previous two. Find the next four terms.

Think · 2 min

Sketch, test a small case, or make a first guess.

Pair · 3 min

Compare methods. Find one thing you agree on and one question.

Be ready to share your route, even if it is unfinished.

Compare before checking

Put two attempts in conversation.

Explain

Tell a partner how you started. Let them finish before responding.

Question

Which step feels certain? Which step still needs evidence?

Build

Combine one useful idea from each attempt into a new route.

Can you explain your partner’s method in your own words?

Board conversation

Notice before naming.

Look for structure

Try a small case and record every step.
Separate what changes from what stays fixed.
State the pattern in a sentence before naming the theorem.

Make a conjecture

We think ___ because ___. A case that might break this is ___.

What changed? What stayed the same?

Useful language

Now the language has a job.

recurrence

rule linking terms

initial condition

starting data

closed form

formula using only the index

Use one of these words to explain something you already noticed.

The core idea

Keep the idea. Explain the conditions.

a_n=a_{n-1}+a_{n-2},\qquad a_0=1,\quad a_1=2

The idea

Define a sequence by how each new term depends on earlier ones.

What to remember

A recurrence is a local construction rule with global consequences.

What information does this method keep?

Worked example

One route, one decision at a time

The problem

Solve $a_{n}$ = a_(n-1) + 3 with $a_{1}$ = 4.

Predict each next step

Each step adds 3.
The sequence is arithmetic.
After n - 1 steps, add 3(n - 1).
Thus $a_{n}$ = 3n + 1.

Which step needs a reason, not just a calculation?

See the structure

The structure behind recurrence relations

a_n=a_{n-1}+a_{n-2},\qquad a_0=1,\quad a_1=2

For n at least 2, count binary strings without consecutive 1s by their final digit.

Read the representation

Define a sequence by how each new term depends on earlier ones.

Look together

Point to the part that explains the relationship. Change one input and predict what changes.

What can you see here that was harder to see in words?

Two approaches

Compare the reasoning.

Structural method

Name the objects and conditions.
Use a reusable relationship.
Explain why the method must work.

Brute force

May help with a small test case.
Can hide the general reason.
Becomes unreliable when the scale grows.

Which route can you justify? What would you test first?

Reasoning clinic

Find the step that needs repair.

Investigate these concerns

May help with a small test case.
Can hide the general reason.
Becomes unreliable when the scale grows.

Repair together

Name the missing condition. Try a small counterexample. Rewrite one step so it is justified.

A useful correction explains why the original step fails.

Work together

Investigate recurrence relations

Your shared task

Groups classify objects by the last move and turn the classification into a recurrence.

Explore

4 min - test cases

Compare

5 min - form a rule

Defend

4 min - defend the rule

Everyone should be able to explain the group’s final claim.

Share → connect

What did the room discover?

Bring your evidence

Share a conjecture, a useful example, or a route that changed.

Connections to investigate

A recurrence is a local construction rule with global consequences.
A complete solution names the condition that makes the method valid.
A counterexample is useful when it exposes a missing condition.

Which observation explains the mechanism?

Try → explain → check

Your turn to build the route

The problem

Find a recurrence and initial conditions for binary strings of length n with no consecutive 1s.

Predict each next step

Let $a_{0}$ = 1 for the empty string and $a_{1}$ = 2 for the strings 0 and 1.
For n ≥ 2, strings ending in 0 follow any valid length n - 1 string.
Those ending in 1 must end in 01 after a valid length n - 2 string.
Thus $a_{n}$ = a_(n-1) + a_(n-2), with $a_{0}$ = 1 and $a_{1}$ = 2.

Which step needs a reason, not just a calculation?

Convince a skeptic

Turn a pattern into a proof.

The claim

A recurrence needs enough initial conditions to determine every later term.

Build the argument

The rule only explains how to move from known terms.
Different starting values can follow the same rule.
Initial conditions choose one sequence from the family.

What does each step rely on? Where are the conditions used?

Stretch together

Take the idea somewhere new.

Transfer challenge

How many ways can a student climb 10 stairs using steps of size 1 or 2?

Choose a route

Name the objects, the conditions, and a relationship that might help. Start with one test case.

What transfers from the earlier example? What is different?

Peer review

Check the argument together.

Restate

Explain the other person’s route before evaluating it.

Test

Try a small case or boundary case. Does every step still make sense?

Revise

Fix an assumption, add a reason, or find a second method.

Where did your thinking change?

Beyond this problem

Where might this idea help?

A connection

Recurrences describe algorithms, populations, tilings, paths, and recursively built structures.

Discuss the model

What objects or quantities are represented?
Which relationship from today is preserved?
What assumption would you check before trusting it?

Invent one more situation with the same structure.

Exit ticket

Leave with an explanation.

Question 1

A sequence starts 2, 5 and each term is the sum of the previous two. Find the next four terms.

Question 2

Explain why this claim is true: A recurrence needs enough initial conditions to determine every later term.

Question 3

How many ways can a student climb 10 stairs using steps of size 1 or 2?

Choose one. Write an answer with a reason, then compare with a partner.

Reflect together

Make your thinking visible.

I used to think…

Name a claim or approach that changed.

Now I can explain…

Give the reason, not only the result.

I still wonder…

Write a question worth bringing to the next circle.

Share one sentence. Ask one follow-up question.

The idea to keep

A recurrence is a local construction rule with global consequences.

Use the matching practice sets to move from direct examples to proof and transfer.

A good question belongs to everyone in the room.