Derive (2), (3), (4), & (5) individually from (1). Derive and explain please!

Assumption: Compound interest. The original equations for (Net) Present value of all streams of even cash flows: CF. CF NPV CFo (1+ r) (1+ r) (1+ r) PMT PMT PMT (1) NPV (PMT) (1+ r) (1+ r) (1+ r) However, we have "short-cut" formulas for cash flow streams with even cash flows of equal amounts: Annuity of N payments in N years happening at the end of each year PMT (2) [(1 r) PV (Annuity) PV (Annuity) PMT or r (1+r) Annuity Due with N payments in N years happening at the beginning of each year or PV(A. Due) PMT (1 r) [(1 r)N 1] (3) PV (A. Due) PMT (1 +r) lr r (1+r) Perpetuity of N payments going into infinity happening at the end of each year PMT (4) PV (Perpetuity) r Perpetuity of N payments going into infinity happening at the beginning of each year PMT (5) PV (Perpetuity Due) (1 r) .(Please make sure you understand definitions of annuity, annuity due, perpetuity, and perpetuity due). Assumption: Compound interest. The original equations for (Net) Present value of all streams of even cash flows: CF. CF NPV CFo (1+ r) (1+ r) (1+ r) PMT PMT PMT (1) NPV (PMT) (1+ r) (1+ r) (1+ r) However, we have "short-cut" formulas for cash flow streams with even cash flows of equal amounts: Annuity of N payments in N years happening at the end of each year PMT (2) [(1 r) PV (Annuity) PV (Annuity) PMT or r (1+r) Annuity Due with N payments in N years happening at the beginning of each year or PV(A. Due) PMT (1 r) [(1 r)N 1] (3) PV (A. Due) PMT (1 +r) lr r (1+r) Perpetuity of N payments going into infinity happening at the end of each year PMT (4) PV (Perpetuity) r Perpetuity of N payments going into infinity happening at the beginning of each year PMT (5) PV (Perpetuity Due) (1 r) .(Please make sure you understand definitions of annuity, annuity due, perpetuity, and perpetuity due)