Problem 55 - 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. $$ a_{n}=n^{2}+5 $$

Step-by-Step Solution

Answer

The given sequence \(a_{n}=n^{2}+5\) is neither an arithmetic sequence nor a geometric sequence.

Step 1: Determine if the sequence is arithmetic

To check if the sequence is arithmetic, calculate the difference between consecutive terms. The formula for an arithmetic sequence is \(a_{n} = a_1 + (n-1)*d\), where \(d\) is the common difference. For this sequence, the difference between two consecutive terms \(a_{n+1} - a_{n}\) gives \((n+1)^2 + 5 - (n^2 + 5)\), which simplifies to \(2n + 1\). Because this is not a constant, implying the difference between consecutive terms are not the same, we can say this sequence is not an arithmetic sequence.

Step 2: Determine if the sequence is geometric

To check if the sequence is geometric, we calculate the ratio between consecutive terms. The formula for a geometric sequence is \(a_{n} = a_1*r^{(n-1)}\), where \(r\) is the common ratio. For this sequence, the ratio of two consecutive terms \(a_{n+1} / a_{n}\) gives \((n+1)^2 + 5 / (n^2 + 5)\). This simplifies to \((n^2 + 2n + 1 + 5) / (n^2 + 5)\), which is not a constant. This implies that the ratio of consecutive terms is not the same, so we can say this sequence is not geometric.

Step 3: Verify if the sequence is neither arithmetic nor geometric

Having established that the sequence does not meet the criteria for being either arithmetic or geometric, it is determined that the sequence is neither an arithmetic sequence nor a geometric sequence.

Key Concepts

Arithmetic Sequence

An arithmetic sequence is a list of numbers in which each term after the first is obtained by adding a constant, called the common difference, to the previous term. The formula defining an arithmetic sequence is:

• \( a_n = a_1 + (n-1) imes d \)

Where:

• \( a_n \) is the \( n^{th} \) term
• \( a_1 \) is the first term
• \( d \) is the common difference

In a simple example like 3, 5, 7, 9, the common difference \( d \) is 2, since each term increases by 2. If the differences between all consecutive terms are not constant, as in the given exercise sequence \( a_n = n^2 + 5 \), the sequence is not arithmetic.

Geometric Sequence

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. The general form of a geometric sequence can be represented as:

• \( a_n = a_1 imes r^{(n-1)} \)

Where:

• \( a_n \) is the \( n^{th} \) term
• \( a_1 \) is the first term
• \( r \) is the common ratio

For instance, in the sequence 2, 4, 8, 16, the common ratio \( r \) is 2, because you multiply each term by 2 to get the next term. If the ratios of consecutive terms in a sequence are not the same, as observed in the given sequence \( a_n = n^2 + 5 \), then the sequence does not qualify as geometric.

Common Difference

The common difference in an arithmetic sequence is the consistent amount added to each term to get to the next. It is crucial because it determines how quickly the sequence progresses. It’s calculated by subtracting any term from the next in the sequence:

• \( d = a_{n+1} - a_n \)

Let's take the sequence 10, 15, 20, 25. Here, the common difference is 5, because each term is obtained by adding 5 to the previous one. In the exercise, when calculating the difference \( (n+1)^2 + 5 - (n^2 + 5) \), the result was \( 2n + 1 \), which is variable. Since it isn't constant, there is no common difference, clarifying why the sequence isn't arithmetic.

Common Ratio

The common ratio in a geometric sequence is the factor by which we multiply each term to get its successor. It can be calculated as follows:

• \( r = \frac{a_{n+1}}{a_n} \)

Knowing the common ratio helps predict and identify future terms in the sequence. For example, the sequence 5, 10, 20, 40 has a common ratio of 2. This means every term is obtained by multiplying the previous one by 2. In the given sequence, calculating the common ratio resulted in \( \frac{(n+1)^2 + 5}{n^2 + 5} \), which was not constant. That is why the sequence didn’t fit the pattern of a geometric sequence, as the ratios fluctuated rather than stayed the same.