Problem 18 - Algebra and Trigonometry

Question

Find the first five terms of the sequence and determine if it is geometric. If it is geometric, find the common ratio and express the \(n\)th term of the sequence in the standard form \(a_{n}=a r^{n-1} .\) $$ a_{n}=4+3^{n} $$

Step-by-Step Solution

Answer

The first five terms are 7, 13, 31, 85, 247. The sequence is not geometric.

Step 1: Calculate the First Term

To find the first term, substitute \( n = 1 \) into the equation: \[ a_1 = 4 + 3^1 = 4 + 3 = 7 \] Thus, \( a_1 = 7 \).

Step 2: Calculate the Second Term

Substitute \( n = 2 \) into the equation:\[ a_2 = 4 + 3^2 = 4 + 9 = 13 \] Thus, \( a_2 = 13 \).

Step 3: Calculate the Third Term

Substitute \( n = 3 \) into the equation:\[ a_3 = 4 + 3^3 = 4 + 27 = 31 \] Thus, \( a_3 = 31 \).

Step 4: Calculate the Fourth Term

Substitute \( n = 4 \) into the equation:\[ a_4 = 4 + 3^4 = 4 + 81 = 85 \] Thus, \( a_4 = 85 \).

Step 5: Calculate the Fifth Term

Substitute \( n = 5 \) into the equation:\[ a_5 = 4 + 3^5 = 4 + 243 = 247 \] Thus, \( a_5 = 247 \).

Step 6: Determine if the Sequence is Geometric

To determine if the sequence is geometric, check if the ratio between consecutive terms is constant. The ratio \( r \) can be computed:\[ r_1 = \frac{a_2}{a_1} = \frac{13}{7} \] \[ r_2 = \frac{a_3}{a_2} = \frac{31}{13} \] Since the ratios are not the same, the sequence is not geometric.

Key Concepts

Sequence Terms

Sequence terms are individual elements that make up a sequence. Think of them as building blocks laid out one after another. In the context of our given exercise, these terms are calculated using a formula that changes depending on the position of the term, denoted by the variable \( n \).
The sequence given by the exercise is defined by the formula \( a_{n}=4+3^{n} \), which means each term is found by plugging in a value of \( n \).
For example, substituting \( n = 1 \) gives us the first term, \( a_1 = 7 \), substituting \( n = 2 \) gives \( a_2 = 13 \), and so forth. You continue this process, changing \( n \) from 1 to 5, to find the first five terms: 7, 13, 31, 85, and 247.
Each of these numbers is known as a sequence term.

• First term: 7
• Second term: 13
• Third term: 31
• Fourth term: 85
• Fifth term: 247

Understanding sequence terms is crucial to grasp larger concepts like finding patterns or establishing if a sequence is arithmetic or geometric.

Common Ratio

The common ratio in a geometric sequence is what you multiply one term by to get to the next term. It's consistent across the relationship between terms in truly geometric sequences.
However, based on the given exercise, we have calculated consecutive ratios and found that the sequence does not consistently provide the same ratio. This inconsistency is highlighted when you compute the ratios:
- For the first two terms: \( r_1 = \frac{13}{7} \)
- For the second and third terms: \( r_2 = \frac{31}{13} \)
Since \( r_1 \) and \( r_2 \) are not the same, this confirms that there is no common ratio, meaning the sequence isn't geometric.
In essence, for a sequence to be geometric, this common ratio must be constant for all term pairs. But in this scenario, it's not. Thus, these ratios don't align, further solidifying that the sequence doesn't truly follow a geometric pattern.

Nth Term Formula

The \(n\)th term formula is a valuable tool in sequences, used to express the term at any given position in a sequence. For geometric sequences, it's given by \(a_n = a\cdot r^{n-1}\), where \(a\) is the first term, \(r\) is the common ratio, and \(n\) represents the term number.
However, since our sequence wasn't geometric due to varying ratios, this specific formula doesn't apply. That said, our sequence still has a specific \(n\)th term formula given by \(a_n = 4 + 3^n\). This formula helps us determine any term's value if we know the position number \(n\).
Such formulas are pivotal because they allow us to jump straight to any term in the sequence without calculating each preceding term.

• The \(n\)th term formula enables a quick resolution to identifying specific sequence terms.
• In geometric terms, consistency in the formula elements is key, but variations in our exercise highlighted its inability to fit a geometric formula.

Understanding how and when to apply these formulas will allow solving complex sequence problems with precision and ease.