residual income

for present value of residual income when do we apply ri/1+r-w versus ri/(1+r-w)*(1-r)^n-1.

will the q explicitly say “terminal value of residual income”? this appears in the Gregory Armishaw Case Scenario where the paragraph says this: Armishaw next presents Exhibit 2, which contains the basis for his estimates for the share price (as of 15 January 2014) if he assumes a terminal value in 2023 arising from treating 2023’s residual income as a perpetuity. Stack questions Armishaw’s assumption in his 2014 valuation (Exhibit 2) that a perpetuity would best describe the terminal value of the stream and suggests that residual income should fade over time. Stack further suggests that a persistence factor of 0.50 might be appropriate.

for stack they apply ri/1+r-w and not ri/(1+r-w)*(1-r)^n-1. why is this?