Problem 9 - Precalculus

Question

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 \(a_{1}=6, r=2\).

Step-by-Step Solution

Answer

The 8th term of the geometric sequence, \(a_{8}\), when the first term, \(a_{1} = 6\), and the common ratio, \(r = 2\), is 768.

Step 1: Understand the problem

The given sequence is a geometric sequence (a sequence where each term is the previous term multiplied by a constant). The first term, \(a_{1}\), is 6, the common ratio, \(r\), is 2 and the term to find, \(a_{8}\), is the 8th term in this sequence.

Step 2: Apply the general formula

To find the nth term of a geometric sequence, we use the formula \(a_{n} = a_{1} \cdot r^{(n-1)}\). Here \(n=8\), \(a_{1}=6\), and \(r=2\). So we substitute these values into our formula to get \(a_{8} = 6 \cdot 2^{(8-1)}\).

Step 3: Simplify the expression

Firstly, calculate the power which is \(2^{(8-1)} = 2^7 = 128\). Then perform the multiplication which gives \(a_{8} = 6 \cdot 128 = 768\).

Key Concepts

Geometric Sequence Formula

Understanding the geometric sequence formula is crucial to solving problems related to geometric sequences. A geometric sequence is a series 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 (r). The formula for the nth term of a geometric sequence is:
\( a_n = a_1 \times r^{(n-1)} \)
where:

• \( a_n \) is the nth term of the sequence,
• \( a_1 \) is the first term,
• \( r \) is the common ratio, and
• \( n \) is the term number.

This formula can easily determine any term in the sequence provided the first term and the common ratio. For example, in our exercise, to find the 8th term (\( a_8 \)) of a sequence with the first term as 6 (\( a_1 \)) and a common ratio of 2 (\( r \)), we would use the formula mentioned above. The power of the common ratio corresponds to one less than the term number because the first term is already accounted for by the \( a_1 \) value.

Common Ratio

The common ratio of a geometric sequence can be considered its defining feature. It is the factor by which we multiply each term to get the next term. To find the common ratio (r), you simply divide any term in the sequence by the previous term—as long as the sequence is geometric, this value will remain constant.Common ratio is a valuable concept because it reflects the rate at which the sequence grows or shrinks. If the common ratio is greater than 1, the terms will increase; if it's between 0 and 1, the terms will decrease; and if the common ratio is negative, the terms will alternate in sign. In our exercise, we're given that the common ratio is 2, which means each subsequent term will be twice as large as the previous one, resulting in exponential growth.

Nth Term Calculation

Calculating the nth term of a geometric sequence is straightforward once the first term and the common ratio are known. Using the geometric sequence formula and plugging in the appropriate values will yield the desired term.For example, in the provided exercise, to calculate the 8th term (\( a_8 \)), we identify our first term \( a_1 = 6 \) and our common ratio \( r = 2 \), and then we realize that \( n = 8 \). Substituting these into our formula, we get:\[ a_8 = a_1 \times r^{(8-1)} \]\[ a_8 = 6 \times 2^{7} \]Finally, we perform the calculations step by step: the exponent first (\( 2^{7} = 128 \)), then the multiplication (\( 6 \times 128 = 768 \)), resulting in the 8th term being 768, as shown in the step by step solution. Understanding this process is essential for correctly solving for the nth term in any geometric sequence.