Since the nature of triangular numbers have been given, I give you the formula for the nth triangular number (let it be t_n)

t_n = n(n + 1)/2

Now, we set up an equation. Since we want to find triangular numbers between 50 and 70, then

50 < t_n < 70

We substitute the formula for t_n

50 < n(n + 1)/2 < 70

We multiply all members of the inequality by 2, to cancel the denominator in the middle.

100 < n(n + 1) < 140

Since n and n + 1 are consecutive integers, then we want to find 2 consecutive integers whose products are between 100 and 140. Since 10 x 11 = 110 and 11 x 12 = 132, then 10 and 11 are what we are looking for.

Thus, the triangular numbers are t_10 and t_11, or

t_10 = 10(10 + 1)/2 = 10(11)/2 = 5(11) = 55

t_11 = 11(11 + 1)/2 = 11(12)/2 = 11(6) = 66

Therefore, the two triangular numbers between 50 and 70 are 55 and 66.

__________________________________

A square is an integer which is an integer multiplied by itself. Let s be the square number. Thus, 4 is a square (since 2 · 2 = 4 and -2 · -2 = 4) and 100 is a square (since 10 · 10 = 100 and -10 · -10 = 100)

Thus, the squares between 300 and 500 are 324, 361, 400, 441, and 484. (18·18, 19·19, 20·20, 21·21, and 22·22, respectively)

^_^_^