Miscellaneous

Elementary
GCD
LCM
Definitions

Percents

Ratios

Square Roots

Reverses

Junior High
Cube Roots

Mult By 9 (#1)
Mult By 9 (#2)

Mult By 8

Polynomials

Positive Integral Divisors

Sum Of Positive Integral Divisors

Relatively Prime 

Triangular Numbers

Remainders

Divide By 40

Divide By 25

Divide By 9

High School

Pentagonal Numbers

Hexagonal Numbers

Heptagonal Numbers

Octagonal Numbers

Manipulating Polygonal Numbers

Dividing Factorials #1

Dividing Factorials #2

Solving Triangular Numbers

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Manipulating Polygonal Numbers:

A.  In working with polygonal numbers there are certain patterns that develop when using successive polygonal numbers:

1.  The difference of 2 successive polygonal numbers is:

(x-2)n - (x-3), where n is the largest x-agonal number

2.  In other words, to find the difference of two successive x-agonal numbers, subtract 2 from x and multiply by n and subtract (x-3).

Ex [1]  The difference of the 6^th and 7^th octagonal numbers is ___.

a.  Since "octa" means 8, the formula is:  (8-2)n - (8-3) or 6n - 5. 

b.  Using the largest number 7, we get 6(7) - 5 or 37.

c.  The answer is 37.

Ex [2]  The difference of the 9^th and 10^th decagonal numbers is ___.

a.  Since "deca" means 10, the formula is:  (10-2)n - (10-3) or 8n - 7.

b.  Using the largest number 10, we get 8(10) - 7 or 73.

c.  The answer is 73.

B.  We can use this above information to do a problem of another type:

Ex [3]  If the 8^th pentagonal number is 92, then the 9^th pentagonal number is __________.

a.  First, find the expression for the difference.

b.  Since "penta" means 5, the formula is: (5-2)n - (5-3) or 3n - 2.

c.  Using the largest number 9, we get 3(9) - 2 or 25. 

d.  If the difference is 25, we can add 25 to 92 to get the 9^th pentagonal number.

e.  92 + 25 = 117.

f.  The answer is 117. 

Ex [4]  If the 40^th octagonal number is 4720, then the 39^th octagonal number is __________.

 a.  The expression for the difference of octagonal numbers is: 6n - 5.

b.  Using the largest number 40, we get 6(40) - 5 or 235.

c.  If the difference is 235, we can subtract 235 from 4720 to find the 39^th octagonal number.

d.  4720 - 235 = 4485.

e.  The answer is 4485.