_

R ^ 2 = 1 - ((n-1)/(n- k - 1)) * (1- (R ^2))

_

R ^ 2 = 1 - ((n-1)/(n- k - 1)) + (n - 1/n - k -1)*(R^2) _

R ^ 2 = (n - k - 1)/(n - k - 1) - ((n-1)/(n - k - 1)) + (n - 1/n - k - 1)*(R^2)

_

R ^ 2 = (n - k - 1 - n + 1)/(n - k - 1) + (n - 1/n- k - 1)*(R ^ 2)

_

R ^ 2 = (-k/n - k -1) + ((n-1)/(n - k - 1))*(R ^2)

_

R ^ 2 = ((n - 1)*(R^2) - k)/(n - k - 1)

_

If R ^2 = 1, then R ^ 2 = R ^ 2

One possible problem with this derivation is converting 1 to n - k - 1/n - k -1

Another problem with the equation is if n = k + 1, in which case the denominator would be 0.

_

If I derive R^2 with the definition, the answer is different:

_

R^2 = 1 - ((n -1)/(n - k - 1))*(1 - R^2)

R ^2 = (Total variation - Unexplained variation)/Total variation

_

R ^ 2 = 1 - ((n-1)/(n-k-1))*((Total variation - Total variation + Unexplained variation)/Total variation)

_

R ^ 2 = 1 - ((n - 1)/(n-k-1))*(Unexplained variation/Total variation)

_

R ^ 2 = (Total variation - ((n- 1)/(n-k-1))*Unexplained variation)/Total variation

Since (n - 1/n - k - 1) > 1 for k >= 0,

_

R ^ 2 > R ^ 2 for k >= 0