at present, there is one-hundred thousand in a trust fund. The fund earns two % interest per year, but ten% of the fund is given away each year. Also, ten-thousand is put into the trust fund per year.
1a) Obtain and solve the difference equation by generating functions
To obtain and solve the difference equation for this scenario using generating functions, we need to understand the various elements involved.
Let's denote the year number by 'n'. The amount in the trust fund at the beginning of year n is given by Tn. According to the given information, the following events happen each year:
1. The fund earns 2% interest per year.
2. 10% of the fund is given away.
3. $10,000 is deposited into the trust fund.
To generate the difference equation, we can use generating functions, which are a useful tool in discrete mathematics.
Let's denote the generating function for the initial amount as T(x), where each term Tn is the coefficient of xn.
First, let's consider the effects of interest and withdrawal. Since 10% of the fund is given away each year, the effective interest rate is 2% - 10% = -8% = -0.08. This means that each term's coefficient should be multiplied by (1 - 0.08x) to account for the loss due to the withdrawal.
Next, we need to account for the $10,000 deposited annually. This adds an additional 10,000 to each term's coefficient.
Thus, the generating function becomes:
T(x) = (1 - 0.08x) * T(x) + 10,000
To solve this difference equation using generating functions, we rearrange the equation to isolate T(x):
T(x) - 0.08xT(x) = 10,000
Factoring out T(x):
T(x) * (1 - 0.08x) = 10,000
Now, dividing both sides by (1 - 0.08x):
T(x) = 10,000 / (1 - 0.08x)
To solve this generating function, we can use partial fraction decomposition. We express T(x) as a sum of partial fractions:
T(x) = A/(1 - 0.08x) + B
To find the values of A and B, we can multiply through by the denominator and equate coefficients:
10,000 = A + B(1 - 0.08x)
Comparing coefficients:
A = 10,000
B(1 - 0.08x) = 0
Since (1 - 0.08x) ≠ 0, B must be zero.
Therefore, the generating function becomes:
T(x) = 10,000 / (1 - 0.08x)
To extract the coefficients and solve the difference equation, we use the Binomial Theorem, which states that:
(1 + a)ⁿ = 1 + n(a) + n(n-1)(a²)/2! + n(n-1)(n-2)(a³)/3! + ...
Applying this to the generating function, we have:
T(x) = 10,000 * (1 + 0.08x + (0.08x)² + (0.08x)³ + ...)
The coefficient of xn in the generating function represents the amount in the trust fund in year n. Therefore, the difference equation can be solved by evaluating the coefficient of xn in the expression:
Coefficient of xn = 10,000 * (0.08)n
To solve for the amount in the trust fund at a specific year n, substitute the value of n into the equation:
Tn = 10,000 * (0.08)n
Thus, the difference equation is Tn = 10,000 * (0.08)n, where Tn represents the amount in the trust fund in year n.