Problem 49 - Precalculus with Limits

Question

In Exercises 47-62, write an expression for the apparent \(n\)th term of the sequence. (Assume that \( n \) begins with 1.) \( 0, 3, 8, 15, 24, \dots \)

Step-by-Step Solution

Answer

The expression for the apparent nth term of the sequence is \(n^2 + n - 2\).

Step 1: Identifying the Differences between Consecutive Terms

Calculate the differences between consecutive terms of the sequence to try and spot a pattern. By subtracting, we get 3 - 0 = 3, 8 - 3 = 5, 15 - 8 = 7 and 24 - 15 = 9. The differences are therefore 3, 5, 7, and 9.

Step 2: Identifying the Pattern in the Differences

See if there's a pattern in the differences between the consecutive terms. Here, the differences themselves form an arithmetic sequence: they increase by 2 each time. This suggests that our initial sequence is a quadratic sequence.

Step 3: Deriving a Formula

For quadratic sequences, the formula is typically of the form: \(an^2 + bn + c\). Given that our sequence starts from zero, the constant term (c) will be zero. Our formula now is \(an^2 + bn + 0\). Considering that the coefficient of \(n^2\) is half of the common difference of the differences in consecutive terms, we have \(a = 2/2 = 1\). Also, considering that each term of the sequence corresponds to \(n^2 + n\), we can see that the coefficient of n is 1. So, the formula is \(n^2 + n\).

Step 4: Verifying the Formula

To verify our formula, we can input some values of \(n\) and see if it produces the correct terms. For example, for \(n = 1\), \(n^2 + n = 1 + 1 = 2\), not 0 as expected. This indicates that we have to modify our formula. From our term formula and the pattern, we can deduce that to obtain the value of each term in the sequence, we need to subtract 2. Therefore, the formula for the nth term of our sequence is \(n^2 + n - 2\).

Key Concepts

Arithmetic Sequences

In an arithmetic sequence, each term after the first is generated by adding a constant number, called the common difference, to the previous term. Imagine lining up a set of numbers where each one is slightly bigger or smaller than the last. This is exactly what an arithmetic sequence looks like.
When you look at the differences between consecutive terms in an arithmetic sequence, you'll find they are always consistent.

• The sequence progresses evenly without wild leaps.
• For example, in the sequence 2, 5, 8, 11, the common difference is 3.
• This simplicity is why arithmetic sequences are often a starting point when analyzing more complex ones.

If a sequence's differences between terms aren't constant but the differences of those differences are (like in the problem we are considering), it suggests a different type of sequence, possibly quadratic.

nth Term Formula

Formulating the \(n\)th term is like writing a recipe for each term using a math formula. It gives a way to find any term in the sequence without listing them all. For quadratic sequences, this formula usually takes the shape \(an^2 + bn + c\).
In our exercise, the sequence's differences formed a pattern that helped us shape its nth term formula.

• We derived the formula by observing the pattern that appeared in the sequence differences.
• The foundational formula for quadratic sequences often begins with \(an^2\) as quadratics are governed by square terms.
• Identifying the coefficients is key, and often involves using the characteristics of the differences.

Remember, verifying the correctness of this formula with initial terms ensures accuracy. It also uncovers any necessary adjustments, as was the case in the given equation \(n^2 + n - 2\).

Polynomial Expressions

Polynomial expressions are composed of several terms, each involving variables raised to whole number powers. Familiar forms like \(ax + b\) are linear polynomials, but they can become as complex as \(an^n + a_{n-1}n^{n-1} + ... + a_0\).
Working with them requires several key strategies:

• Identifying how each term contributes to the overall structure.
• Simplifying where possible, especially when creating or solving equations.
• Understanding the degree of the polynomial, which guides its curve behavior on a graph.

In our case, quadratic expressions like \(n^2 + n - 2\) withheld a form requiring us to discern polynomial components (coefficients and constants) from sequence analyses. Recognizing and applying these concepts is vital in sequences, especially when transitioning from linear (arithmetic) to more intricate quadratic relationships.