High School

The pentagonal numbers are whole numbers that are represented by pentagonal shapes. The first four pentagonal numbers are shown.

a. Complete the following table and describe the pattern in the Number of Dots column.

b. Make a sketch to represent the fifth pentagonal number.

c. How many dots will be in the ninth pentagonal number?

d. Is there a pentagonal number that has 200 dots in its shape? If so, which one?

e. Write a formula for the number of dots in the nth pentagonal number.

Answer :

Final answer:

The number of dots in the nth pentagonal number is given by the formula (3n^2-n)/2.

Explanation:

The pattern for pentagonal numbers is that each number is the sum of the number before it and a sequence of odd numbers (n-1)*2, where n is the position in the pentagonal number sequence. To sketch the fifth pentagonal number, you would draw a pentagon with dots at each vertex and filling in the shape such that each side contains 5 dots. The ninth pentagonal number would have 117 dots since the formula for pentagonal numbers is (3n^2-n)/2 where n is the position in sequence. The result would be (3*9^2-9)/2=117. The closest pentagonal number to 200 is the 18th pentagonal number with 171 dots (the 19th has 228 dots, which is more than 200). The formula for the nth pentagonal number mentioned earlier is (3n^2-n)/2

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