2^x+1 + 2^x+2 + ..... +2^x +40 = 2^43 - 8

To solve this equation, we can use the formula for the sum of a geometric series:

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

In this case, the first term (a) is 2^(x+1), the common ratio (r) is 2, and the number of terms (n) is 40.

So, the equation becomes:

S = 2^(x+1)(1 - 2^40) / (1 - 2)

Simplifying further:

S = 2^(x+1)(1 - 2^40) / (-1)

Now, let's simplify the right side of the equation:

2^43 - 8 = 2^43 - 2^3 = 2^43 - 2^3(1) = 2^43 - 2^3(2^40)

Comparing this to the equation, we can see that:

S = 2^43 - 2^3(2^40)

Therefore, the equation is true for any value of x.