Paul is talking to Pesi on the phone.

He is trying to describe a pattern to Pesi but he can’t find the words.

If Paul’s pattern is: 3, 7, 11, 15, …how can he describe to Pesi how to get any member of the number pattern?

How can Paul tell Pesi how to get the 50th number in as simple a way as possible?

In this problem students must describe a sequence in words, and state a rule for any term in the sequence, using an expression for the nth term of a sequence.

Other related Level 3 problems include: Toothpick Squares and Race To 100

### Problem

Paul is talking to Pesi on the phone. He is trying to describe a pattern to Pesi but he can’t find the words.

If Paul’s pattern is:

3, 7, 11, 15, …how can he describe to Pesi how to get any member of the number pattern?

How can Paul tell Pesi how to get the 50th number in as simple a way as possible?

### Teaching sequence

- Introduce the lesson with a game of "who belongs". In this game the students try to guess members of a number sequence. As they guess the teacher sorts the numbers into 2 lists on the board – "does belong" and "doesn't belong".

As the lists develop the students attempt to guess the rule – however they keep this rule secret until the end of the game when rules are shared.

For example: The rule is multiples of 4

Do belong 12, 24, 8, 4

Do not belong 1, 0, 5, 7, 6, 13, 15 - Pose the problem to the class.
- As the students work ask them questions that require them to describe the patterns in their own words.
- If the students are having problems with finding the patterns encourage them to explore the size of the "jumps" between the numbers.
- As the students work to find the 50
^{th}number remind them to find the easiest way. Although the students could continue the sequence 50 times this is time consuming and does not require them to find a rule for any term in the pattern. - Share descriptions of the number patterns.

#### Extension to the problem

Pesi has a pattern. It is 3, 6, 12, 24, …

How can Pesi describe to Paul how to get any member of the number pattern? How can Pesi tell Paul how to get the 50th number in as simple a way as possible?

### Solution to the problem

Paul’s sequence is 3, 7, 11, 15, …

The first term is 3. From there Paul keeps adding 4. Paul can say "Pesi, you start at 3 and keep adding 4. That way you’ll get all members of my pattern."

The 50th term can be found by ‘adding 4’ until Pesi gets to the 50th term.

The more efficient way is to tell Pesi, "You just take 3 and add 49 4s." To which Pesi replies "Great, so the 50th term is 3 + (49 x 4) = 199."

#### Solution to the extension

Pesi's sequence is 3, 6, 12, 24, …

He starts with 3 and doubles each time to get the next number in the sequence. He says, "Paul, take 3 and keep doubling."

Since doubling is done one less time than the number of the term in the sequence, he tells Paul, "Take 3 and double it, then double it again and keep doing this for 49 doublings. So the 50th term is 3 x 2 x 2 x … x 2, where there are 49 2s." A calculator may be needed to work that one out. The number 3 x 2^{49} is very big! It’s roughly 3 with 15 zeros after it!