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Junior High
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Mult By 9 (#1)
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High School

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Dividing Factorials #1

Dividing Factorials #2

Solving Triangular Numbers

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Solving x² - y² = a³, Where x And y Are Triangular Numbers

A.  In this type of problem we are squaring triangular numbers which means (see Triangular Numbers):

B.  However, this is the same formula for adding 1³ + 2³ + ... + n³ (see Adding Cubes).

C.  Therefore, x² - y² = a³ means:

(1³ + 2³ +...+ x³) - (1³ + 2³ +...+ y³) = a³

The only way for this to be true is if x = a and y = a - 1

D.  So, to solve this problem, choose x = (a)^th triangular number and choose y = (a - 1)^th triangular number.

E.  Examples:

Ex [1]  x² - y² = 6³, x and y are negative triangular numbers, then x = ______.

a.  For this problem we need x = 6^th triangular number and y = 5^th triangular number.

b.  Since we are only concerned with x, we need the 6^th triangular number which is 21.  Since the answer has to be negative, the answer is -21.

c.  If the question had asked for y, the answer would have been the 5^th triangular number which is 15.  Since the answer has to be negative, it would be -15.

Ex [2]  x² - y² = 512 and x,y are triangular numbers, then y = ____.

a.  For this problem you should know that 512 = 8³.  So x is equal to the 8^th triangular number and y is equal to the 7^th triangular number.

b.  The answer is the 7^th triangular number or 28.

c.  If the question had asked for x, the answer would be the 8^th triangular number or 36.