Question

Government Expenditures A government program that currently costs taxpayers $$\$ 2.5$$ billion per year is cut back by 20 percent per year. (a) Write an expression for the amount budgeted for this program after \(n\) years. (b) Compute the budgets for the first 4 years. (c) Determine the convergence or divergence of the sequence of reduced budgets. If the sequence converges, find its limit.

Short answer

The budgeted amount for the program after \(n\) years is given by the expression \(2.5 * (0.8)^n\) billion dollars. The budgets for the first four years are \$2.0 billion, \$1.6 billion, \$1.28 billion, and \$1.024 billion. The budget amount will decrease each year and eventually converge to \$0 billion.

Step-by-step solution

Formulate the expression

The government program is cut back by 20 percent per year, that means it retains 80% (or 0.8) of its value every year. So after one year, the budget would be \(0.8 * 2.5\) billion dollars, and in general, after \(n\) years, the amount budgeted (\(a_n\)) would be \(a_n = 2.5 * (0.8)^n\) billion dollars.

Compute the budgets for the first 4 years

Substitute \(n = 1, 2, 3, 4\) into the expression \(a_n = 2.5 * (0.8)^n\). You will find that the budgets for the first four years are approximately \$2.0, \$1.6, \$1.28, and \$1.024 billion respectively.

Determine the convergence or divergence

The budget is a geometric sequence with common ratio \(0.8\) which is less than \(1\) and greater than \(-1\), so the sequence of reduced budgets converges. Find its limit by taking the limit as \(n\) goes to infinity of \(a_n = 2.5 * (0.8)^n\), which is equal to 0. So the limit of the sequence is \$0 billion.

Key concepts

Convergence and Divergence

Understanding whether a sequence converges or diverges is crucial in calculus, particularly when dealing with sequences that describe real-world phenomena like government budgets or population growth.

A sequence converges if the terms approach a specific value as the number of terms goes towards infinity. Think of it as zooming in on a target: if the numbers in the sequence get closer and closer to a point without actually reaching it, the sequence is converging on that point. In our example with the government budget cuts, since each year's budget is 80% of the previous year's, each number in this sequence is getting closer to zero. This means the sequence is converging to zero.

Visualizing Convergence

Imagine watching a balloon deflate very slowly; with each breath of air that escapes, the balloon gets slightly smaller. Eventually, it'll be almost flat - never fully deflated since we're doing it slowly, but it gets so close we can say it has 'converged' to being flat.

On the other hand, if the terms do not settle near any one value, we say the sequence diverges. Like throwing a ball in an open field, it keeps going further away without settling down. In terms that relate to our budget example, if the cuts were increasing the budget each year instead of decreasing it, the budget numbers would go higher and higher, potentially leading to a divergent sequence.

Limit of a Sequence

The limit of a sequence is a fundamental concept in calculus representing the value that the terms of a sequence 'approach' as the number of terms increases indefinitely. It's a bit like predicting where a runner will eventually reach if they keep slowing down at a consistent rate - eventually, they'll come to a stop, and that stopping point is the 'limit' of their distance over time.

For the sequence of government budget cuts, the limit as the number of years increases is zero dollars. This is because each subsequent budget is 80% of the previous year, and 80% repeatedly taken from any positive amount will get closer and closer to zero without ever being negative. This diminishing nature brings us to an interesting point in sequences: if a sequence converges, it has a limit.

Calculating the Limit

To calculate the limit of the sequence of budgets, we assess what happens to our formula, \(2.5 \times (0.8)^n\), as \(n\) becomes very large. As \(n\) approaches infinity, \((0.8)^n\) approaches 0 since any fraction smaller than 1 raised to an infinite power dwindles to 0. Consequently, multiplying this by any number, like 2.5 in our example, will also result in 0. Hence, the limit of the budget sequence is $0.

Expressing Sequences Algebraically

To express sequences algebraically means writing them down in a compact, mathematical form using equations. This is not just for neatness; it provides us a powerful way to analyze and manipulate sequences, which can be applied to finding patterns, making predictions, or solving complex problems.

In the government budget example, the algebraic expression for the sequence that represents the yearly budget after \(n\) years is \(a_n = 2.5 \times (0.8)^n\). This is named a geometric sequence because each term is found by multiplying the previous term by a constant called the 'common ratio'. In this case, the common ratio is 0.8. Knowing this allows us to predict future budgets and calculate their limits.

Importance of Algebraic Expression

By converting the sequence into an algebraic expression, we create a user-friendly, universal language that can be understood and utilized across various mathematics fields. It tells us at a glance how the values in the sequence are related and how they will develop over time. Students who master expressing sequences algebraically can tackle a wide range of problems in calculus and beyond.