Problem 9 - College Algebra

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, a1, and common ratio, r. Find \(a_{8}\) when \(a_{1}=6, r=2.\)

Step-by-Step Solution

Answer

The value of the 8th term, given \(a_{1}=6\) and \(r=2\), is 768.

Step 1: Substitute the given values into the formula

The formula for the nth term of a geometric sequence is \(a_{n}=a_{1}*r^{(n-1)}\). Given \(a_{1}=6\), \(r=2\), and \(n=8\), substitute these values into the formula to get \(a_{8}=6*2^{(8-1)}\).

Step 2: Simplify Exponent

First, subtract 1 from 8 to find the exponent, which results in 7. So, the formula becomes \(a_{8}=6*2^{7}\).

Step 3: Calculate \(2^{7}\)

Next, calculate \(2^{7}=128\).

Step 4: Multiply \(a_{1}\) by \(2^{7}\)

Now, simply multiply the first term \(a_{1}\) by the resulting value of \(2^{7}\), thus we get \(a_{8}=6*128\)

Step 5: Solve for \(a_{8}\)

Multiplying 6 by 128 gives 768. Therefore, \(a_{8}=768\) given the first term 6 and a common ratio of 2.

Key Concepts

Nth Term Formula

In a geometric sequence, finding any term is easy with the nth term formula. This formula is given by: \[ a_n = a_1 \cdot r^{(n-1)} \] Here:

• \(a_n\) is the term you want to find.
• \(a_1\) is the first term of the sequence.
• \(r\) is the common ratio.
• \(n\) is the term number you want.

By simply plugging in these values, you'll know exactly how to find any term in the sequence. The formula uses exponentiation to simplify the process, so make sure you are comfortable with exponents.

Common Ratio

The common ratio is a key concept in geometric sequences. It is the factor by which we multiply each term to get the next term in the sequence. In this exercise:

• The common ratio, \(r\), is 2.

Understanding the common ratio:

• If \(r > 1\), the sequence grows.
• If \(0 < r < 1\), the sequence shrinks.
• If \(r = 1\), all terms are equal.
• If \(r < 0\), the terms alternate in sign.

The common ratio helps us understand the pattern of the sequence and predicts subsequent terms efficiently. In our example, multiplying by 2 rapidly increases each term's value.

Exponentiation

Exponentiation is a mathematical operation that raises a number (the base) to the power of an exponent. In the context of geometric sequences, exponentiation helps us quickly calculate terms far into the sequence. Let's see how it works in this exercise:

• We had to calculate \(2^{7}\).
• The result, \(128\), is obtained by multiplying 2 by itself 6 more times: 2, 4, 8, 16, 32, 64, 128.

Notice how quickly the numbers grow, emphasizing the power of using exponents in calculations. In a sequence with a large common ratio or for larger term numbers, without exponentiation, this would be cumbersome.

Arithmetical Sequences

While geometric sequences use multiplication to find each subsequent term, arithmetical sequences rely on repeated addition. It's important to recognize the difference between these two types of sequences. In an arithmetic sequence:

• A constant, known as the "common difference," is added to each term to find the next term.
• For example, in the sequence 2, 4, 6, 8, the common difference is 2.

In contrast, our exercise focuses on a geometric sequence, where the use of a common ratio and exponentiation means the terms grow or shrink exponentially, rather than linearly as in arithmetic sequences.