The first four triangular numbers are:
The difference between the first and second triangular numbers is 2.
The difference between the second and third triangular numbers is 3.
The difference between the third and fourth triangular numbers is 4.
All the triangular numbers follow this pattern.
The difference between the r th triangular number and the (r+3 )^th triangular number is 126.
What is the value of r ?

Answer:

16) B

Let difference between r and r + 1 = M
Let difference between r + 1 and r + 2 = N
Let difference between r + 2 and r + 3 = O

The sum of the difference = 126
M + N + O = 126

By understanding the example given in the question, we can infer that:
N = M + 1
O = N + 1

Since N = M + 1,
O = M + 1 + 1 = M + 2
M + N + O = 126
M + M + 1 + M + 2 = 126
3M + 3 = 126
3M = 123
M = 41

M is the difference between r^th and (r + 1)^th triangular number.
From the example described in the question, we know that:
Difference = initial triangular number + 1.

E.g. Between 2^nd triangular number and 3^rd triangular number, difference = 3
Initial triangular number is the 2^nd triangular number.

Therefore,
M = (r^th triangular number) + 1
41 – 1 = r^th triangular number
r^th triangular number = 40

So the answer is B.