The sum of n numbers of geometric progression is GP=(2^n+1)-1. Find the first term and the common difference.

To find the first term (a) and the common difference (d) of a geometric progression, we need to use the formula for the sum of a geometric series:

Sn = a(1 - r^n)/(1 - r)

Where:
- Sn denotes the sum of the first n terms of the geometric progression
- a is the first term of the geometric progression
- r is the common ratio of the geometric progression

In this case, we are given the sum of n terms as GP = (2^(n+1)) - 1.

Setting this equal to Sn in the formula, we have:

(2^(n+1)) - 1 = a(1 - r^n)/(1 - r)

We need to simplify this equation to isolate a and r, the first term and the common ratio respectively.

Step 1: Expand the equation

(2^(n+1)) - 1 = a - a * r^n - r^(n+1) + r^n

Step 2: Rearrange the equation

a(1 - r^n) = (2^(n+1)) - 1 + r^(n+1) - r^n

Step 3: Divide both sides by (1 - r^n)

a = [(2^(n+1)) - 1 + r^(n+1) - r^n] / (1 - r^n)

Now, we can find the values of a (first term) and r (common ratio) by substituting the given expression for the sum, GP=(2^(n+1))-1, into the equation.

a = [(2^(n+1)) - 1 + r^(n+1) - r^n] / (1 - r^n)

Finally, we have an equation for the first term (a) and common ratio (r) in terms of n, but finding their exact values depends on having additional information or constraints related to the problem.