Kathryn deposits 100 into an account at the beginning of each 4-year period for 40 years

Let the 4-year effective rate be r. Then, we have

r = [(1+i)^4]-1

Future Value (FV) of annuity due = (PMT/r)*[(1+r)^n - 1]*(1+r) where PMT = 4-year deposit of 100; r = 4-year effective rate; n = number of deposits made

For the 40-year period, we have: PMT = 100; n = 40/4 = 10

FV after 40 years (FV40) = (100/r)*[(1+r)^10 -1]*(1+r)

Similarly, for the 20-year period, we have: PMT = 100; n = 20/4 = 5

FV after 20 years (FV20) = (100/r)*[(1+r)^5 - 1]*(1+r)

It is given that 5*FV20 = FV 40, so we have

5*(100/r)*[(1+r)^5 - 1]*(1+r) = (100/r)*[(1+r)^10 -1]*(1+r)

(500/r)*[(1+r)^5 - 1] = (100/r)*[(1+r)^10 - 1]

5*[(1+r)^5 - 1] = [(1+r)^10 - 1]

Let (1+r)^5 be denoted by x. Then, we have

5*(x - 1) = x^2 - 1

x^2 - 5x + 4 = 0

This is a quadratic equation which can be solved as:

x^2 -4x -x + 4 = 0

x*(x-4) -1*(x-4) = 0

(x-1)*(x-4) = 0

So, x = 1 or x = 4

Solving for r, if x = 1, we get: r = [x^(1/5)] -1 = [1^(1/5)]-1 = 1 - 1 = 0. Since r cannot be zero, this implies that x has to be 4.

So, r = [4^(1/5)]-1 = 31.95%

FV40 = (100/r)*[(1+r)^10 -1]*(1+r) = (100/31.95%)*[(1+31.95%)^10 -1]*(1+31.95%)

= 312.98*15.00*1.32 = 6,194.72

X = 6,194.72 (Answer)