Geometric Progression:
A geometric progression is a sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. It can be represented as P, P * R, P * R^2, P * R^3, and so on.

Product of the Terms:
To find the product of the terms Tn, Tn+1, Tn+2, ..., Tn+m-1, we need to express each term in terms of the first term P and the common ratio R.

Let's assume that the first term is P and the common ratio is R. Therefore, the terms of the geometric progression can be represented as:

Tn = P * R^(n-1)
Tn+1 = P * R^n
Tn+2 = P * R^(n+1)
Tn+3 = P * R^(n+2)
.
.
.
Tn+m-1 = P * R^(n+m-2)

Product of the Terms:
To find the product of the terms Tn, Tn+1, Tn+2, ..., Tn+m-1, we can multiply all the terms together:

Tn * Tn+1 * Tn+2 * ... * Tn+m-1 = (P * R^(n-1)) * (P * R^n) * (P * R^(n+1)) * ... * (P * R^(n+m-2))

Using the properties of exponents, we can simplify the expression:

Tn * Tn+1 * Tn+2 * ... * Tn+m-1 = P * P * P * ... * P * R^(n-1) * R^n * R^(n+1) * ... * R^(n+m-2)

Since P is multiplied m times and R is multiplied m-1 times, the product can be written as:

Tn * Tn+1 * Tn+2 * ... * Tn+m-1 = P^m * R^((n-1) + n + (n+1) + ... + (n+m-2))

The exponent of R can be simplified:

Tn * Tn+1 * Tn+2 * ... * Tn+m-1 = P^m * R^(m(n-1) + m(n) + m(n+1) + ... + m(n+m-2))

The sum inside the exponent can be simplified as well:

Tn * Tn+1 * Tn+2 * ... * Tn+m-1 = P^m * R^(m(n-1 + n + (n+1) + ... + (n+m-2)))

The sum inside the exponent is an arithmetic series with n terms, starting from n-1 and ending at n+m-2. The sum of an arithmetic series can be calculated using the formula:

sum = (n/2) * (first term + last term)

In this case, the first term is n-1 and the last term is n+m-2:

sum = (n/2) * (n-1 + n+m-2)
sum = (n/2) * (2n+m-3)

Substituting back into the expression:

Tn