The future-sum form of the geometric gradient series is used for inflation-adjusted retirement savings / investment plan.  The cash flow diagram is shown in blue to the left.  The equation is as follows:

The entire inflation-adjusted savings series is calculated from A₁ as follows:

The entire inflation-adjusted retirement series is calculated from R₁ as follows:

If, in pre-retirement planning, we have an initial lump sum, B_O, at time j=0, we will need to convert it to an equivalent amount B_m, at time j=m, as shown in the cash flow diagram to the left, as follows:

In pre-retirement planning, we want to work with current-year estimates of our annual retirement need, R_O, in today's dollars (at time j=1).
We then calculate R₁ from R_O as follows:

For pre-retirement planning, we must connect the above three equation sets at the point j=m, to be coincident with the point k=0, and set the equality relation as follows: 

The Consolidated Pre-retirement Planning Equation

Note that the the consolidated equation is in terms of  R_O, and A₁ , which are terms stated in current dollars at time j=1 , and B_O stated in dollars which are still essentially current dollars from j=0.  This gives you solutions in terms you can easily relate to.

Three pre-retirement planning solutions can be generated form the consolidated equation:

• Solution 1: Solving for A₁, given R_O, i, r, and both m and n (which means we must also know D_O, D_R , and D_D)

• Solution 2:  Solving for R_O, given A₁, i, r, and both m and n (which means we must also know D_O, D_R , and D_D)
   

• Solution 3: Solving for D_R, given A₁, R_O,  i, r, and (m+n) (which means we must also know D_O and D_D, but not m or n)

(See example to see how this is done)

From D_O, D_R , and D_D we calculate m and n:

m = D_R - D_O
 n = D_D  - D_R