Problem 50 - Algebra 2

Question

Find the 10 th term of each geometric sequence. $$ a_{9}=-5, r=-\frac{1}{2} $$

Step-by-Step Solution

Answer

The 10th term of the sequence is 2.5

Step 1: Understand the Geometric Sequence Formula

The formula for the nth term in a geometric sequence is \(a_{n} = a_{1} \times r^{(n-1)}\). But in this case, the provided term is not the first. We instead have the 9th term, which means we have \(a_{9} = a_{1} \times r^{8}\), and we need to find the 10th term.

Step 2: Use Given Values to Calculate the 10th Term

The 10th term, \(a_{10}\), can be found by multiplying the 9th term, \(a_{9}\), by the common ratio \(r\). So, \(a_{10} = a_{9} \times r = -5 \times (-\frac{1}{2})\).

Key Concepts

Understanding the Common Ratio

In a geometric sequence, the common ratio is a key concept. It's the constant factor that each term is multiplied by to get the next term. To find it, you take any term in the sequence and divide it by the term right before it.

For our exercise, the given common ratio is \(-\frac{1}{2}\). This means each term in the sequence is half the size of the previous term, but with a sign change.

• This is important because it dictates how the sequence grows or shrinks at each step.
• The sign of the common ratio will also affect the sign of the terms in your sequence.

The nth Term Formula

The nth term formula of a geometric sequence allows us to find any term without listing all the preceding ones. The formula is: \[ a_{n} = a_{1} \times r^{(n-1)} \]where:

• \(a_{n}\) is the nth term you're trying to find,
• \(a_{1}\) is the first term of the sequence,
• \(r\) is the common ratio,
• \(n\) is the term number you're interested in.

In the given exercise, though we start with \(a_{9}\), we eventually use this formula to find the 10th term. Thus, it highlights the flexibility of the formula in identifying any term based on the sequence pattern.

Geometric Progression of the Sequence

A geometric progression is simply another name for a geometric sequence. This progression exhibits multiplicative changes, where you multiply each term by a constant, the common ratio, to get the next term.

Imagine starting with a grain of rice, and multiplying the count over several iterations. A geometric progression like this can dramatically increase or decrease terms, depending on the values:

• If the common ratio is greater than one, terms will grow larger at each step.
• If it's less than one (but positive), terms shrink.
• For negative common ratios, like our example of \(-\frac{1}{2}\), the sequence will alternate signs while reducing in magnitude.

Steps of Sequence Calculation

Calculating terms of a geometric sequence requires understanding and applying several strategic steps. For calculating specific terms:

1. Identify the known values: your starting term and the common ratio.
2. Use the relationship between terms to progress through the sequence.

In our example, we directly used the 9th term \(a_{9} = -5\) and multiplied it by the common ratio \(r = -\frac{1}{2}\) to find the next term, \(a_{10}\). Thus: \[ a_{10} = a_{9} \times r = -5 \times -\frac{1}{2} = 2.5\]These steps highlight how sequence calculations allow us to step forward in a progression, emphasizing the practicality of a formula-based approach over manual computation.