**Middle School Number Sense Lesson 64: Pentagonal Numbers**

Today's lesson continues our discussion of

**polygonal numbers**--building on the idea of triangular numbers (

**TriangularNum**) from last week. Questions about

**pentagonal numbers**appeared

**7 times**last year, with the median placement at

**question # 64**.

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**Definition:**

A number is a

**pentagonal number**if it represents the number of objects that can form a

**regular pentagon**. The first several pentagonal numbers are 1, 5, 12, 22, 35, 51, 70, .... I recommend that at least this many pentagonal numbers be memorized.

**How to Solve:**

- If you are asked to give the
**nth**pentagonal number, use the formula**n(3n-1)/2**. - There is a special relationship between pentagonal numbers and triangular numbers. The
**nth pentagonal number**is 1/3 of the**(3n - 1)th triangular number**.

**Example 1: The fourth pentagonal number is ___**

- Use the formula
**n(3n-1)/2**. - 4[3(4) - 1]/2 = 4(12 - 1)/2 = 4(11)/2 = 44/2 =
**22**.

**Example 2: The 7th pentagonal number is ___**

- Use the formula
**n(3n-1)/2**. - 7[3(7) - 1]/2 = 7(21 - 1)/2 = 7(20)/2 = 7(10) =
**70**.

**Example 3: The difference between the 8th & 9th pentagonal numbers is ___**

- Use the formula
**n(3n-1)/2**. **8th**: 8(3 x 8 - 1)/2 = 4(23) =**92**.**9th**: 9(3 x 9 - 1)/2 = 9(26)/2 = 9(13) =**117**.- 117 - 92 =
**25**.

**Example 4: The sum of the first 3 pentagonal numbers is ___**

- I recommend memorization here, but they can be calculated if necessary.
- 1 + 5 + 12 =
**18**. : The sum of the first**Note****n**pentagonal numbers happens to be the product of**n**and the**nth triangular number**.

**Example 5: The sum of the 5th pentagonal number and the 5th triangular number is ___**

- If you don't have these memorized, you can calculate them.
- 15 + 35 =
**50**. : This type of question also forms a specific pattern. The answer will always be**Note****2n(n)**.

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

**Up Next for Middle School: Prism**