The sum of two consecutive terms of a progression is 237 while the difference between those terms is 5. If the first term of the progression is 11 and the difference between two consecutive terms of the progression is the same, then what are the respective positions of those two consecutive terms?

To solve this problem, we need to use the properties of an arithmetic progression (AP). In an AP, the difference between any two consecutive terms is constant.

Given that the first term (a) is 11 and the common difference (d) is 5, we can express the nth term of the AP as a + (n-1)d.

We are told that the sum of two consecutive terms is 237 and the difference between those terms is 5. Let's denote these two terms as Tn and Tn+1.

So, we have:

Tn + Tn+1 = 237 and Tn+1 - Tn = 5.

Substituting the expressions for Tn and Tn+1 from the formula of an AP, we get:

[a + (n-1)d] + [a + nd] = 237 and [a + nd] - [a + (n-1)d] = 5.

Solving these two equations, we get:

2a + 2nd - d = 237 and d = 5.

Substituting the given values a = 11 and d = 5, we get:

211 + 2n5 - 5 = 237 and 5 = 5.

Solving for n, we get n = 23.

So, the two consecutive terms are the 23rd and 24th terms of the progression.