| VRⁿ - Vₙ = D₁ | Rⁿ - 1 |
| R - 1 |
Solving for D₁; we have:
| (4) D₁ = | (VRⁿ - Vₙ)(R - 1) | = | (VRⁿ - Vₙ)r |
| Rⁿ - 1 | Rⁿ - 1 | ||
| = periodic depreciation charge | |||
That this is the same as the amount of the periodic depreciation charge by the sinking fund method plus interest on the investment, is seen by comparing formula (4) with formula (3). Formula (4) may be written as
| (VRⁿ - Vₙ)r + Vr - Vr |
| Rⁿ - 1 |
i.e., the quantity Vr-Vr = 0 is put into the numerator. Adding zero (Vr - Vr = 0) cannot change its value. Performing the multiplication indicated by the parentheses, we have
| VRⁿr - Vₙr + Vr - Vr |
| Rⁿ - 1 |
Rearranging the terms, we have
| (Vr - Vₙr) + (VRⁿr - Vr) |
| Rⁿ - 1 |
which factored in each group becomes