Question In sequence of triangular numbers find the followin

Question: In sequence of triangular numbers, find the following. (a) Two triangular numbers whose sum and difference are also triangular numbers. (b) Three successive triangular numbers whose product is a perfect square. (c) Three successive triangular numbers whose sum is a perfect square.

Solution

Triangular numbers are like :

1 , 3 , 6 , 10 , 15 , 21 , 28 , 36 , 45 , 55 , 66 , 78 , 91 , 105 , 120

a) Two triangular numbers whose sum and difference are also triangular numbers :

15 and 21.
21 - 15 --> 6, also a triangular number
21 + 15 --> 36, which is also a triangular number

b) Three successive triangular numbers whose product is a perfect square :

From the list above, 6 , 10 and 15 are the numbers.
Reason : 6*10*15 = 900 ---> 30^2, i.e a perfect square

c) Three successive triangular numbers whose sum is a perfect square :

15 , 21 and 28 are the numbers
Reason : 15 + 21 + 28 = 36 + 28 = 64 ---> 8^2, a perfect square