Q2 VALUATION, DEPRECIATION AND THE RATE-BASE 
For each dollar invested the second year: 
a(x+d2+a(z+é)3+ --- + adollars 
| HE + 2)» — 1] dollars. 
For each dollar invested the nth year: a dollars. 
Therefore the total amount S, in the replacement fund at 
the end of the nth year, after deducting the $1 replacement 
requirement of that year: 
S. = Zatti (14+) t—14+---+(@+13) —1]—1 (1) 
S,= 42a = Dl —nf x (2) 
So =2l@ +a — @ +4) — mi] — 1. (3) 
There will be in the replacement fund for each dollar annually 
invested: 
At the end of the (# + 1)st year: 
Spi1 = Sn (1 +1) + na — 1. 4) 
At the end of the (n + 2)d year: 
Sez = S, (1 +12)2 + (ma — 1) (1 +14) + na — 1. (5) 
At the end of the (# + 3)d year: 
Seis =S, (1 +128 + (na — 1) (1 +7) 
+ (na —1) (1414) +na—1, (6) 
and so on; and at the end of the mth year: 
Sm =S, (1 +3) + (na — 1) (1 +2)! 
+ ma —1)(@ +24 os +e —1. (7) 
Substituting the value of S, and summarizing the series: 
a ; Nis ; Rossin 
Sr =5[( + dyn — (1 + Om — i (1 + 5)" 
— +d + Ea +r — dma — 1), ©) 
or 
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