PentagonalNum

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 ___

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

Example 2: The 7th pentagonal number is ___

1. Use the formula n(3n-1)/2.
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 ___

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

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

1. I recommend memorization here, but they can be calculated if necessary.
2. 1 + 5 + 12 = 18.
3. Note: The sum of the first 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 ___

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

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

Up Next for Middle School: Prism