Problem 10 - Algebra and Trigonometry

Question

In Exercises \(9-16,\) use the formula for the general term (the nth term) of a geometric sequence to find the indicated term of each sequence with the given first term, \(a_{1},\) and common ratio, \(r .\) Find \(a_{8}\) when \(a_{1}=5, r=3\)

Step-by-Step Solution

Answer

After solving, we find that \(a_{8} = 5 \times 2187 = 10935.\) Therefore, the 8th term of the sequence is 10935.

Step 1: Identify knowns

First, we identify the knowns: the first term, \(a_{1}=5,\) and the common ratio, \(r=3.\)

Step 2: Apply the formula

Next, we use the formula for the nth term of a geometric sequence: \(a_{n}=a_{1}\times r^{n-1}\). By substituting the values of \(a_{1}=5, r=3\), and \(n=8\), we get \(a_{8}=5 \times 3^{8-1}\).

Step 3: Calculate the 8th term

We can simplify \(3^{8-1}\) to \(3^{7}\) which equals 2187. Therefore, \(a_{8}=5 \times 2187\).

Key Concepts

nth term

The "nth term" of a sequence is a way to express any term in a sequence using its position number. In a geometric sequence, which involves multiplying by a consistent number, or "common ratio," each time, the nth term can be calculated using a specific formula. This is useful when you want to find a term without listing all the previous terms.
For the nth term of a geometric sequence, the formula is: \(a_{n}=a_{1} \times r^{n-1}\). Here, \(a_{n}\) represents the term you're looking for, \(a_{1}\) is the first term, \(r\) references the common ratio, and \(n\) is the position number of the term in the sequence.
In the exercise, to find \(a_{8}\), you use this formula and plug in the values for \(n\), which is 8, and the other known values: \(a_{1}=5\) and \(r=3\). It helps make complex calculations simpler by giving a direct path to the desired term.

first term

The "first term" in a geometric sequence is the starting point of the sequence. It is commonly denoted as \(a_{1}\). This term is crucial because it acts as a foundation for calculating all subsequent terms in the sequence.
Unlike arithmetic sequences that add or subtract a constant, a geometric sequence multiplies by a constant—the common ratio. So, knowing the first term allows you to initiate those multiplicative steps. For example, in the sequence provided, \(a_{1}\) is given as 5.
Understanding the first term helps determine both the direction and specific values the sequence will take. It's like setting the ball rolling; from this initial term, everything else is generated through multiplication by the common ratio.

common ratio

The "common ratio" in a geometric sequence is the factor by which we multiply to get from one term to the next. It is constant throughout the sequence, ensuring that the sequence progresses by the same multiplicative amount each time.
This factor is symbolized as \(r\), and it can be any real number, positive or negative. A positive common ratio maintains the same sign as the starting term, causing the sequence terms to grow or shrink incrementally. A negative ratio will cause alternating signs between consecutive sequence terms.
In the exercise example, \(r=3\), meaning each term is three times the previous one. Knowing the common ratio helps predict the behavior of the sequence; whether it grows, shrinks, or oscillates, and significantly influences the calculation of the nth term through the sequence formula.

sequence formula

The "sequence formula" for a geometric sequence is a powerful tool allowing the determination of any term in the sequence without having to list every previous term. The formula is expressed as \(a_{n} = a_{1} \times r^{n-1}\), where \(a_{n}\) is the term of interest, \(a_{1}\) the first term, \(r\) the common ratio, and \(n\) the term position.
This formula stems from the sequential application of the common ratio, implying repeated multiplication starting from the first term. By inputting these values, you can calculate any term's value directly, which is particularly useful in sequences with many terms or very large term positions.
For instance, finding the 8th term \(a_{8}\) in the exercise uses this exact formula; plugged with \(a_{1}=5\), \(r=3\), and \(n=8\), it effortlessly leads to the result by calculating \(5 \times 3^{7}\). This makes handling even seemingly complex sequences manageable and straightforward.