Reconsider Parts (a) through (f) of Problem 2. For each "true" statement, develop a mathematical proof based on the time value of money factor equations from Table 2.6 in Chapter 2.

Data from problem 2

Determine whether each of the following is true or false. In each case, assume \(P\) is located at \(t=0\) and \(F\) is located at \(t=n\), and the \(A\) s are spread uniformly over the planning horizon.

a. \((A \mid P i \%, 1)=(F \mid P i \%, 1)\)

b. \((A \mid P i \%, n)=(F \mid P i \%, n)\); for \(n>1\)

c. \(P(A \mid P i \%, n)=P(F \mid P i \%, n)(A \mid F i \%, n)\)

d. \(F(A \mid F i \%, n)=F(P \mid F i \%, n)(A \mid P i \%, n)\)

e. \(P(A \mid P i \%, n)=P(A \mid P i \%, n-3)+P(A \mid P i \%, 3)\); for \(n>3\)

f. \(P(A \mid P i \%, n)=P(A \mid P i \%, n-3)(A \mid P i \%, 3)\); for \(n>3\)

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TABLE 2.6 Summary of Discrete Compounding Interest Factors. To Find Given Factor Symbol Name P F (1+i)" (P|Fi%,n) Single sum, present worth factor F P (1+i)" (F\Pi%,n) Single sum, compound amount factor (1+i)" 1 P A (P|Ai%,n) Uniform series, present worth factor i(1+i)" i(1 + i)" A P (A|Pi%,n) Uniform series, capital recovery factor (1+i)" -1 (1+i)" 1 F A (F|Ai%,n) Uniform series, compound amount factor i i A F (1+i)"-1 (A|Fi%,n) Uniform series, sinking fund factor [1-(1+ni)(1 + i)"] P G (P|G1%,n) Gradient series, present worth factor A (1+i)" - (1+ni) [(1+i)"-1] (A|Gi%,n) Gradient to uniform series conversion factor P Aj F Aj 1 (1 + j)" (1 + i) i-j (1+i)" (1 + j)" i-j for ij (PA, 1%, j%,n) Geometric series, present worth factor forij (FA 1%,j%) Geometric series, future worth factor