Question

The annual yield of wheat in a certain country has been found to equal the average of the yield in the previous two years. If the yields in 1990 and 1991 were 10 and 12 million tons respectively, find a formula for the yield \(k\) years after \(1990 .\) What is the long-term average yield?

Short answer

The long-term average yield is 11 million tons.

Step-by-step solution

Understanding the Problem

We are given the annual wheat yield for 1990 and 1991, which are 10 and 12 million tons, respectively. The yield for any subsequent year is the average of the yields from the two previous years. We need to find a formula for the yield \(k\) years after 1990.

Recursive Formula Identification

We recognize that the yield in any year \(n\), denoted as \(Y_n\), is given by the average of the yields from the previous two years: \(Y_n = \frac{Y_{n-1} + Y_{n-2}}{2}\).

Initial Values Setup

We have the initial yields: \(Y_{1990} = 10\) and \(Y_{1991} = 12\). These will be used to calculate further yields using the recursive formula.

Calculating Further Yields

Using the recursive formula: \(Y_{1992} = \frac{Y_{1991} + Y_{1990}}{2} = \frac{12 + 10}{2} = 11\). Similarly, \(Y_{1993} = \frac{Y_{1992} + Y_{1991}}{2} = \frac{11 + 12}{2} = 11.5\). Continuing this process helps illustrate the pattern.

Analyzing Long-term Behavior

Notice that the difference between consecutive yields is diminishing as it is the average. This means the sequence approaches an equilibrium where consecutive yields are equal. Calculate using the equilibrium condition: \(Y_n = Y_{n-1} = x\), leading to \(x = \frac{x + x}{2} = 11\).

Writing the General Formula

Since each yield depends on averaging two previous ones and stabilizes over time, write the formula as \(Y_{1990+k} = \frac{10 + 12}{2} = 11\) for large \(k\).

Conclusion

The yield approaches 11 million tons when \(k\) is large, and this reflects the long-term average yield.

Key concepts

Recursive Formulas

Recursive formulas are essential tools in mathematical modeling. They allow us to express quantities based on previous terms in a sequence, providing a step-by-step method to calculate future values. For example, in our problem, we look at the annual yield of wheat in a particular country. The yield for each year is determined by averaging the yields of the previous two years.
This is a classic setup for a recursive formula. We represent the yield for year \( n \) as follows:

• \( Y_n = \frac{Y_{n-1} + Y_{n-2}}{2} \)

This formula tells us that to get the yield of year \( n \), we must first know the yields of the two preceding years. Recursive formulas often involve initial conditions which anchor the sequence. In this example, the initial yields are given as 10 and 12 million tons for 1990 and 1991. These values serve as starting points for the recursive process. By applying the recursive formula, we can determine the yields for any subsequent years by repeatedly using the yields already calculated.

Equilibrium States

In recursive sequences, especially those involving a stable process like averaging, we frequently encounter a concept known as an equilibrium state. An equilibrium state is a condition where successive terms of a sequence become stable and no longer change significantly. In our example, as we apply the recursive formula over the years, we notice the differences in yield decrease.
Ultimately, the yield reaches a point where the changes are negligible, indicating that the sequence has entered an equilibrium state. Mathematically, when the sequence reaches this state, the yields become constant, meaning:

• \( Y_n = Y_{n-1} = x \)

To find this constant \( x \), we set the recursive formula equal to each other since each term equals its preceding terms:

• \( x = \frac{x + x}{2} = 11 \)

Hence, the yield stabilizes at 11 million tons. Identifying equilibrium states is crucial, as it helps determine the long-term behavior of the sequence. When sequences stabilize, it simplifies the prediction of future values.

Average Calculation

A key part of this problem involves calculating the average, as averages play a vital role in both the recursive formula and finding the equilibrium state. Calculating an average is a straightforward yet powerful tool in mathematical analysis. In this problem, the average serves to determine the yield of wheat for any given year.
The basic formula for calculating an average of two numbers \( a \) and \( b \) is:

• \( \text{Average} = \frac{a + b}{2} \)

This simple operation is repeated to generate new yearly yields. For example, for 1992, we calculate:

• \( Y_{1992} = \frac{Y_{1991} + Y_{1990}}{2} = \frac{12 + 10}{2} = 11 \)

By understanding how averages work, we can apply this operation to handle recursive sequences efficiently and predict the long-term average yield, which signifies the equilibrium we discussed. When a series averages out like this, it is indicative of an inherent stability or balance in the system.