An n-year annuity-due will pay t² at the beginning of year t for t=1,2,...,n (1 for year 1, 4 for year 2, 9 for year 3, etc.). The effective annual rate of interest is fixed at i. (a) Derive a formula for the present value PV(n) of this annuity at time 0 as a function of the term n and the discount factor v = (1 + i)¹ only. [4] Hint: You may use the following equalities: m m-1 m m Σ(t+1)²v² = [(t+1)²v²+¹ + 2[tv² + [v² 1=0 1=0 1=1 1=0 Note: The formula should NOT include sums or products of the form n n Σx₁ = x₁ + x₂ + + x₂ or x₁x₁x₂xn j=1 j=1 For example, to express ä as a function of n and v, the right answer is 1-v ä not a= [w=1+v+v² +...+v²-1 9 1-v j=0 (b) Verify the formula of PV(n) in part (a) for n=1 and n=2. (c) Prove the formula of PV (n) in part (a) by mathematical induction on n.