Problem 11 - 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 \(a_{12}\) when \(a_{1}=5, r=-2.\)

Step-by-Step Solution

Answer

The twelfth term of the geometric sequence is -10240.

Step 1: Identify the known values

Here, the first term (\(a_{1}\)) is 5, the common ratio (r) is -2 and the term we have to find is the 12th term.

Step 2: Apply the geometric sequence formula

Insert the known values into the formula for the nth term of a geometric sequence, that is, \(a_{n} = a_{1} \cdot r^{(n-1)}\). Therefore, we get: \(a_{12} = 5 \cdot (-2)^{12-1}\).

Step 3: Calculate \(a_{12}\)

After substituting the values into the equation, simplify the expression to get the 12th term of the geometric sequence, which gives: \(a_{12} = 5 \cdot (-2)^{11} = 5 \cdot -2048 = -10240.\)

Key Concepts

General Term

When studying geometric sequences, one key component is understanding the general term. The general term, often referred to as the nth term, is crucial for determining any term in the sequence without listing all the previous terms. In mathematical language, the nth term of a geometric sequence is expressed by the formula: \( a_{n} = a_{1} \cdot r^{(n-1)} \), where:

• \( a_{n} \) is the nth term you want to find,
• \( a_{1} \) is the first term of the sequence, and
• \( r \) is the common ratio.

This formula enables you to jump directly to the desired term without repeated multiplication. This saves time and minimizes the risk of mistakes common in manual calculations. Additionally, understanding the structure of this formula highlights how sequences grow exponentially, explaining why changes in the common ratio can drastically alter the sequence dynamics.

Common Ratio

In a geometric sequence, the common ratio is a constant value that each term is multiplied by to obtain the next term in the sequence. It is a central feature in defining the pattern of the sequence.

• If the common ratio is positive and greater than one, the sequence will increase rapidly.
• If the common ratio is a negative number, the sequence will alternate in sign, meaning it will switch between positive and negative.
• If the common ratio is a fraction between 0 and 1, the sequence will decrease, bringing the terms closer to zero.

To identify the common ratio, simply divide any term in the sequence by the one preceding it, i.e., \( r = \frac{a_{n}}{a_{n-1}} \). This consistency in the multiplicative factor is what defines a sequence as geometric, and understanding it helps in predicting how the sequence evolves.

Nth Term Calculation

Calculating the nth term in a geometric sequence is straightforward once the general term formula is understood. Here's how it works step-by-step:1. **Identify Known Values:** - Determine \( a_{1} \) (the first term of the sequence), - Find \( r \) (the common ratio), - Identify \( n \), which is the position of the term you want to find.
2. **Use the Formula:** - Insert these values into the formula \( a_{n} = a_{1} \cdot r^{(n-1)} \).
3. **Perform the Calculation:** - With the values in place, carry out the exponentiation \( r^{(n-1)} \), and then multiply by \( a_{1} \).
For example, in a sequence where \( a_{1} = 5 \) and \( r = -2 \), to find \( a_{12} \), you'd calculate \( a_{12} = 5 \cdot (-2)^{11} = -10240 \). This step-by-step process will give you the term you need without listing all previous terms, providing both efficiency and accuracy in computation.