Problem 56 - Algebra and Trigonometry

Question

The general term of a sequence is given. Determine whether the sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio. $$ a_{n}=n^{2}-3 $$

Step-by-Step Solution

Answer

The sequence \(a_{n}=n^{2}-3\) is neither an arithmetic sequence nor a geometric sequence.

Step 1: Identify the sequence type

The formula for the nth term of an arithmetic sequence is \(a + (n - 1) * d\) and for a geometric sequence is \(a * r^{(n - 1)}\). Taking a look at the sequence \(a_{n} = n^{2}-3\), we observe that neither of these formulas resembles the given one, since the power on \(n\) in our sequence is \(2\), which makes it a quadratic expression. Therefore, this sequence is neither arithmetic nor geometric.

Step 2: Confirm by checking the differences and ratios

We can verify this by finding differences and ratios for a few terms. For \(n=1,2,3\) we have sequences \(1^2-3=-2\), \(2^2-3=1\), \(3^2-3=6\). The differences are \(1-(-2)= 3\) and \(6-1 =5\), which are not equal. The ratios are \(1/(-2) = -0.5\) and \(6/1=6\), which are also not equal. Hence the sequence is neither arithmetic nor geometric.

Key Concepts

Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This difference is known as the common difference, denoted as \(d\).
Imagine you have a list of numbers where each number is added by the same amount to get the next one. For example, the sequence \(2, 5, 8, 11, \ldots\) has a common difference of \(3\) because you add \(3\) each time to reach the next number.

• General Form: The nth term of an arithmetic sequence can be expressed as \(a_n = a + (n - 1) \cdot d\), where \(a\) is the first term, and \(d\) is the common difference.
• Properties: Every pair of successive numbers has the same difference \(d\).
• Examples: Given \(a = 2\) and \(d = 3\), the sequence is \(2, 5, 8, 11, \ldots\)

Arithmetic sequences are straightforward and predictable. When you spot a pattern where each number is evenly spaced from the last, you're likely dealing with an arithmetic sequence.

Geometric Sequence

A geometric sequence is characterized by each term being a consistent multiple of the previous one. The constant ratio, called the common ratio, is denoted as \(r\).
For instance, consider the sequence \(3, 6, 12, 24, \ldots\). Each term is multiplied by \(2\) to get the next term, so the common ratio is \(2\).

• General Form: The nth term in a geometric sequence can be written as \(a_n = a \cdot r^{(n-1)}\), where \(a\) is the first term and \(r\) is the common ratio.
• Properties: Multiplying each term by \(r\) gives the next term in the sequence.
• Examples: With \(a = 3\) and \(r = 2\), the sequence becomes \(3, 6, 12, 24, \ldots\)

Geometric sequences grow quickly if the common ratio is greater than \(1\), offering expansive or contracting patterns when the ratio is less than \(1\).

Quadratic Sequences

Quadratic sequences are more complex, where the difference itself changes and forms a sequence. In these, the nth term involves squared terms, leading to a parabolic pattern. Unlike arithmetic or geometric, there's no single "common" factor.
Consider the sequence \(1, 4, 9, 16, \ldots\), which stems from the squares of \(1, 2, 3, 4, \ldots\). Here, terms progress as the square of their position: \(n^2\).

• Characteristics: The nth term is typically given as \(a_n = an^2 + bn + c\).
• Behavior: The differences between terms change progressively. For example, differences between \(1, 4, 9, 16\) are \(3, 5, 7\), themselves forming an arithmetic sequence.
• Example: With \(a_n = n^2\), each term represents a perfect square.

Quadratic sequences appear in many natural and engineered settings, representing growth that accelerates or decelerates with each term shift, unlike the linear (arithmetic) or proportional (geometric) sequences.