triangle numbers and square numbers?

Question

investigate and prove that
any square number can be diveded into two triangle numbers

Answer 1

Triangular numbers are
1+2+3+4+5+6+7+8+9+10+11+12+13+14+15 etc
1, 3,6,10,15,21,28,36,45,55,66,78,91,105,120,etc
Square numbers are
1,4,9,16,25,36,49,64,81,100,121,144,169,196,etc

Only true of triangular numbers that are next to each other, eg 1, 3, 6, 9,

If you mess around with triangular numbers, for a while, you may notice that the sum of any two consecutive triangular numbers seems to be a square. 1+3=4, 3+6=9, 6+10=16, etc. Do you think that is is always true? On the right, I have rearranged the dots in two consecutive triangular numbers. Forget for a moment that these are the 5th and 6th triangular numbers, as this works for any two consecutive triangular numbers. This arrangement of dots is indeed a square, and is for any pair of consecutive triangular numbers. In fact the nth triangular number plus the (n+1)th triangular number is (n+1)2.

Answer 2

Triangular Numbers
1 = 1

1 + 2 = 3


1 + 2 + 3 = 6


1 + 2 + 3 + 4 = 10


1 + 2 + 3 + 4 + 5 = 15


1 + 2 + 3 + 4 + 5 + 6 = 21


1 + 2 + 3 + 4 + 5 + 6 + 7 = 28


1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 36


1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45


1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 55

Square Numbers

4, 9, 16, 25, 36, 49, 64, 81, 100


A square number can be divided into 2 triangular numbers

square number is first, followed by the two triangular numbers
The triangular numbers are consecutive (beside each other)
4 is the square number; 3 and 1 are the triangular numbers
4 is 3,1
9 is 3,6
16 is 6,10
25 is 10, 15
36 is 15, 21
49 is 21, 28
etc.
You do the rest ... look for the pattern.

Answer 3

what are triangle numbers? if you mean the Pythagorean theorem then you use two sides of a right triangle labeled a and b to get the hypotenuse by using the equation a^2 +b^2 = c^2 when c = the hypotenuse. otherwise you are speaking gibberish

Answer 4

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