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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