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John invests 1000 in a fund which earns interest during the first year at a nominal rate of K compounded quarterly. During the 2nd year the fund earns interest at a nominal discount rate of K compounded quarterly. At the end of the 2nd year, the fund has accumulated to 1173.54. Calculate K.

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Final answer:

To calculate the nominal rate K for John's investment, we solve the two-stage compound interest formula for the amount accumulated over two years with different compounding methods, leading to a transcendental equation which is solved numerically.

Step-by-step explanation:

We are given that John invests $1000 which grows to $1173.54 over two years through two different interest computation regimes. In the first year, the interest is earned at the nominal rate K compounded quarterly, and in the second year, the fund earns interest at a nominal discount rate of K also compounded quarterly. To find K, we need to work through two stages of calculation, one for each year.

For the first year, the formula for compound interest is:

A = P(1 + \(r/n\))^(nt)

where:
A = the amount of money accumulated after n years, including interest.
P = the principal amount (the initial amount of money).
r = the annual interest rate (decimal).
n = the number of times that interest is compounded per year.
t = the time the money is invested for in years.

So for the first year, we can write:
A1 = 1000*(1+K/4)^(4*1)

For the second year, as the interest is earned at a nominal discount rate, the formula changes slightly, but we will still use the principle of quarterly compounding:

A2 = A1*(1-K/4)^(4*1)

Finally, we are given A2 which accumulates to $1173.54 at the end of the second year, hence:

1173.54 = 1000*(1+K/4)^(4*1)*(1-K/4)^(4*1)

We solve this equation for K using algebraic methods or numerical approximation techniques (such as Newton-Raphson or bisection method) to find the nominal rate K.

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