Question

The difference of the squares of two consecutive triangular numbers is always a _______.

• A. Perfect square
• B. Perfect cube
• C. Prime number
• D. Negative number.

Answer 1

The correct option is B Perfect cube

Triangular numbers are derived by the addition of consecutive natural numbers.
1+2 = 3,
3+3 =6
6+4 =10 and so on.
The triangular number sequence is 1, 3, 6, 10, 15, 21.... and so on.
The squares of the triangular number sequence will be
1. 9, 36, 100, 225 and so on..
Let us find the difference between the two consective squares:
9-1 = 8 = $2^3$ , a perfect cube.
36 - 9 = 27 = $3^3$ , a perfect cube.
100 - 36 = 64 = $4^3$ , a perfect cube and so on.

Hence, the difference of the squares of two consecutive triangular numbers is always a perfect cube.