Recall the definitions of the actuarial notations. Prove the following identities: (a) an i (ai VI-v ). (b) v + 2v2 + 3v3 + + (m - 1)xn1+ no + (m - 1)x+1+(x 2)x7+2 + aolla = + 2v2n2 + v2n-1 (c) 1 (Ia)a a d 1 (d) + + + 1 > 1+r1 $0,1 $1,2 fin-1,n] 1 (1+r2) (1 + rn)" (1 + rn)" where rk is the k-th spot rate and f(k-1,k] is the annual forward rate between time k 1 and k. Recall the definitions of the actuarial notations. Prove the following identities: (a) an i (ai VI-v ). (b) v + 2v2 + 3v3 + + (m - 1)xn1+ no + (m - 1)x+1+(x 2)x7+2 + aolla = + 2v2n2 + v2n-1 (c) 1 (Ia)a a d 1 (d) + + + 1 > 1+r1 $0,1 $1,2 fin-1,n] 1 (1+r2) (1 + rn)" (1 + rn)" where rk is the k-th spot rate and f(k-1,k] is the annual forward rate between time k 1 and k