Problem 56 - College Algebra

Question

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 arithmetic nor geometric, so there are no common differences or ratios to calculate.

Step 1: Identify the Sequence Type

Observe the formula \(a_{n}=n^{2}-3\). An arithmetic sequence has a general formula of \(a_{n}=a_{1} + (n-1)d\) where \(d\) is the common difference. A geometric sequence has a general formula of \(a_{n}=a_{1}r^{(n-1)}\) where \(r\) is the common ratio. Neither of these formulas matches \(a_{n}=n^{2}-3\). Thus, this sequence is neither arithmetic nor geometric

Step 2: Find the common difference or common ratio

As this sequence is neither arithmetic nor geometric, there are no common differences or common ratios to compute.

Key Concepts

Arithmetic Sequence

An arithmetic sequence is a list of numbers where each term is obtained by adding a fixed value to the previous term. This fixed value is known as the common difference, represented by the letter 'd'. The general form of an arithmetic sequence can be described as follows: for the n-th term, the equation is \( a_n = a_1 + (n-1)d \),where \( a_1 \) is the first term and 'n' is the term number in the sequence. For instance, in the sequence 3, 5, 7, 9, ..., the common difference is 2, because each term increases by 2. One can easily predict any term in the sequence by applying the general formula and knowing the first term and the common difference. It is a fundamental concept that forms the basis for various mathematical calculations and problem solving.

Geometric Sequence

Conversely, a geometric sequence is a series of numbers where each term after the first is found by multiplying the previous term by a fixed non-zero number called the common ratio (denoted by 'r'). The formula for the n-th term of a geometric sequence is \( a_n = a_1 \cdot r^{(n-1)} \),where \( a_1 \) is the first term. If we take a sequence like 2, 6, 18, 54, ..., each term is multiplied by 3 to get the next term, which makes the common ratio 3. The concept of the geometric sequence is essential in understanding exponential growth or decay, financial calculations involving compound interest, and many other areas of both mathematics and real-world applications.

Common Difference

The common difference is a key element in an arithmetic sequence. It's the constant value that you add to each term to get the next one in the sequence. This value can be positive, negative, or even zero, leading to sequences that increase, decrease, or remain constant, respectively. As an example, for the arithmetic sequence 4, 7, 10, 13, ... the common difference is 3 because each number is 3 more than the number before it. If we were looking for the fifth term, we could simply add the common difference four times to the first term, which in this case would result in the fifth term being 16. Understanding the common difference is critical for solving problems that involve arithmetic sequences.

Common Ratio

The common ratio, on the other hand, is a defining feature of a geometric sequence. It's the ratio between successive terms in the sequence, and just like the common difference, it remains constant throughout the sequence. In the geometric sequence 3, 9, 27, 81, ..., you can obtain the common ratio by dividing any term by the preceding term (for instance, 9/3 or 27/9), which gives a common ratio of 3. Understanding the common ratio allows us to find any term in the sequence, predict patterns of growth or decline, and it's particularly important for applications related to geometric sequences like in financial models predicting market trends or population studies.