Triangular Numbers and Square Numbers?

Question

What is the connection between the nth triangular number and the nth square number?

Answer 1

The nth triangular number is the sum of the first n integers:
1, 1+2=3, 1+2+3=6, 1+2+3+4=10, etc.
It can be proved that the nth triangular number is equal to n(n+1)/2. Call this Tn. Then 2Tn = n(n+1) = n^2 + n, so the n'th square number is equal to twice the nth triangular number minus n.

Answer 2

Let the square numbers be S_1, S_2, S_3, ... S_n

Where S_n = n²

Let the triangular numbers be T_1, T_2, T_3, .... T_n

Where T_n = 1 + 2 + 3 + .... + n = n/2(n + 1)

Then S_n = T_n + T_(n - 1)

Proof
T_n = n/2(n + 1) = n²/2 + n/2

T_(n - 1) = (n - 1)/2 * ((n - 1) + 1) = n/2(n - 1) = n²/2 - n/2

So T_n + T_(n - 1) = (n²/2 + n/2) + (n²/2 - n/2)
= n²/2 + n/2 + n²/2 - n/2
= n²
= S_n .... QED

Example:
Square numbers are {1, 4, 9, 16, 25, 36, 49, 64, .......}

Triangular numbers are {1, 3, 6, 10, 15, 21, 28, 36, ......}

Sum af 1st two Triangular numbers is 4 = S_2

T_5 + T_6 = 15 + 21 = 36 = S_6 etc The sum of any two consecutive triangular numbers results in a square number.

Answer 3

4 is kind of triangular. 5 is more square. (this part is kind of advanced, but try to follow: 8 is round)