COSMO · STUDY + TEACHING DECK

Sequences

April 18, 2026

A question worth sharing

Generating Functions

Store an entire sequence as coefficients of one formal power series.

ThinkCompareExplain

Today’s route

A question becomes an explanation.

Try

Enter through a problem. A partial attempt counts.

Connect

Use generating function, coefficient extraction, formal series 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

What sequence is encoded by 1/(1-x)?

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.

generating function

series whose coefficients encode a sequence

coefficient extraction

reading [ $x^{n}$ ]

formal series

algebraic object without convergence concerns

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

The core idea

Keep the idea. Explain the conditions.

[x^n]\left(A(x)B(x)\right)=\sum_{i=0}^{n}a_i b_{n-i}

The idea

Store an entire sequence as coefficients of one formal power series.

What to remember

A sequence becomes easier to manipulate when its terms become coefficients.

What information does this method keep?

Worked example

One route, one decision at a time

The problem

Find the generating function for 0,1,2,3,... .

Predict each next step

Start with 1/(1-x) = sum $x^{n}$ .
Differentiate to get 1/(1-x)^2 = sum n $x^{n-1}$ .
Multiply by x.
The result is x/(1-x)^2.

Which step needs a reason, not just a calculation?

See the structure

The structure behind generating functions

[x^n]\left(A(x)B(x)\right)=\sum_{i=0}^{n}a_i b_{n-i}

For A(x) and B(x) with coefficients a and b, multiplication combines all index pairs summing to n.

Read the representation

Store an entire sequence as coefficients of one formal power series.

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 generating functions

Your shared task

Students encode simple choice rules as factors, multiply truncated series, and interpret coefficients.

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 sequence becomes easier to manipulate when its terms become coefficients.
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

Use generating functions to count unordered ways to make n using any number of 1s and 2s.

Predict each next step

Any number of 1s contributes 1/(1-x).
Any number of 2s contributes 1/(1- $x^{2}$ ).
Multiply the two count choices; order is ignored.
The coefficient of $x^{n}$ in 1/((1-x)(1- $x^{2}$ )) is floor(n/2) + 1.

Which step needs a reason, not just a calculation?

Convince a skeptic

Turn a pattern into a proof.

The claim

Multiplying generating functions performs convolution of their coefficient sequences.

Build the argument

The coefficient of $x^{n}$ in a product comes from index pairs i+j=n.
Each pair contributes $a_{i}$ $b_{j}$ .
Summing these products is exactly convolution.

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

Stretch together

Take the idea somewhere new.

Transfer challenge

Find the coefficient of $x^{10}$ in 1/((1-x)(1- $x^{3}$ )).

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

Generating functions solve recurrences, count restricted objects, and connect algebra with combinatorics.

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

What sequence is encoded by 1/(1-x)?

Question 2

Explain why this claim is true: Multiplying generating functions performs convolution of their coefficient sequences.

Question 3

Find the coefficient of $x^{10}$ in 1/((1-x)(1- $x^{3}$ )).

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 sequence becomes easier to manipulate when its terms become coefficients.

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

A good question belongs to everyone in the room.