**Middle School Number Sense Lesson 63: Triangular Numbers**

Today's lesson introduces us into the world of

**polygonal numbers**, which as a group, show up quite often on number sense tests. This year, triangular numbers are the most commonly tested of the polygonal numbers. These questions appeared

**5 times**last year, with a median placement at

**question # 64**.

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**Definition:**

A number is

**triangular**if it represents the number of stacked objects that can form an

**equilateral**

**triangle**, as shown below. The first several triangular numbers are 1, 3, 6, 10, 15, 21, .... Here's

**1:**

**3:**

O O

**6:**

O O

O O O

**10:**

O O

O O O

O O O O

**15:**

O O

O O O

O O O O

O O O O O

**21:**

O O

O O O

O O O O

O O O O O

O O O O O O

**nth triangular number**is the

**sum of the positive integers**up to (and including)

**n**:

**1**st (1) =

**1**

**2**nd (3) = 1 +

**2**

**3**rd (6) = 1 + 2 +

**3**

**4**th (10) = 1 + 2 + 3 +

**4**

**5**th (15) = 1 + 2 + 3 + 4 +

**5**

**6**th (21) = 1 + 2 + 3 + 4 + 5 +

**6**

and so on.

**How to Solve:**

- If you are asked to give the
**nth**triangular number, use the formula:**n(n+1)/2**. - If you are asked to give the
**sum**of two consecutive triangular numbers, simply**square the larger number**. - If you are asked to give the
**difference**between two consecutive triangular numbers, simply**write the****larger number**. - If you are asked for anything else, GOOD LUCK! (Just kidding--you can probably use some combination of these ideas).
*It may be a good idea to memorize the first 6 or so triangular numbers, for quick recall.*

**Example 1: The 5th triangular number is ___**

- Use
**n(n+1)/2**. - 5(5+1)/2 = 5(6)/2 = 30/2 =
**15**.

**Example 2: The 9th triangular number is ___**

- Use
**n(n+1)/2**. - 9(9+1)/2 = 9(10)/2 = 90/2 =
**4****5**.

**Example 3: Which of the following is a triangular number: 5, 15, or 25?**

- If you have the first several triangular numbers memorized, this one will be easy.
- If you don't, try doubling each of the numbers. Check to see if the result can be the product of two consecutive numbers--essentially reverse-engineering our formula of
**n(n+1)/2**. - The answer is
**15**.

**Example 4: The sum of the 10th & 11th triangular numbers is ___**

- Simply
**square**the larger number. - 11 x 11 =
**121**.

**Example 5: The difference between the 25th & 26th triangular numbers is ___**

- Simply
**write**the larger number. - The answer is
**26**.

**Example 6: 36 is the nth triangular number. n = ___**

- Use the idea in Example 3 (step 2). Double 36 to get
**72**. - Now think of two consecutive numbers whose product is 72. They are
**8 & 9**. - The answer is the smaller of the two:
**8**.

**Check back soon for a free worksheet to help you practice TriangularNum.**

**Up Next for Middle School: PentagonalNum**