Problem 15 - A Graphical Approach to Precalculus with Limits

Question

CHECKING ANALYTIC SKILLS Find \(a_{5}\) and \(a_{n}\) for each geometric sequence. Do not use a calculator. $$10,-5, \frac{5}{2},-\frac{5}{4}, \dots$$

Step-by-Step Solution

Answer

\(a_5 = \frac{5}{8}\) and \(a_n = 10 \cdot \left(-\frac{1}{2}\right)^{n-1}\).

Step 1: Identify the First Term

The first term of a geometric sequence is denoted by \(a_1\). In this sequence, \(a_1 = 10\).

Step 2: Determine the Common Ratio

To find the common ratio \(r\), divide the second term by the first term. Here, the second term is \(-5\), and the first is \(10\): \( r = \frac{-5}{10} = -\frac{1}{2} \).

Step 3: Use the Formula for Any Term in the Sequence

The general formula for the \(n\)-th term of a geometric sequence is given by \(a_n = a_1 \cdot r^{n-1}\).

Step 4: Calculate the Fifth Term

Using the formula and the values found, calculate \(a_5\):\[a_5 = 10 \cdot \left(-\frac{1}{2}\right)^{4} = 10 \cdot \frac{1}{16} = \frac{10}{16} = \frac{5}{8}\]

Step 5: Write the Expression for the General Term

Substitute \(a_1\) and \(r\) into the general term formula to find \(a_n\):\[a_n = 10 \cdot \left(-\frac{1}{2}\right)^{n-1}\]

Key Concepts

Common Ratio

In a geometric sequence, each term is obtained by multiplying the previous term by a fixed number, known as the common ratio. The common ratio is a crucial element that defines how the sequence progresses. To find the common ratio in any sequence, you divide one term by the preceding term. For example, in our given sequence, the first term is 10 and the second term is \(-5\).

• The first term (\(a_1\)) is 10.
• The second term is \(-5\).
• Hence, the common ratio (\(r\)) can be calculated as follows:

\[ r = \frac{-5}{10} = -\frac{1}{2}\]This ratio, \(-\frac{1}{2}\), tells us that each term in the sequence is produced by multiplying the previous term by \(-\frac{1}{2}\). Understanding the common ratio allows us to predict and calculate further terms in the sequence with consistent multiplication.

General Term Formula

The general term formula in a geometric sequence helps us find any term without listing all previous terms. Understanding this formula is essential for working with sequences efficiently. For a geometric sequence, the general term \(a_n\) is given by:\[a_n = a_1 \cdot r^{n-1}\]

• \(a_1\) is the first term of the sequence. In our example, this is 10.
• \(r\) is the common ratio. For our problem, it is \(-\frac{1}{2}\).
• \(n\) is the term position you wish to find.

The expression \(r^{n-1}\) allows us to calculate how the factor (the common ratio) applies repeatedly up to the \(n\)-th term. Plugging in the values for both the first term and the common ratio, the general term for our sequence becomes:\[a_n = 10 \cdot \left(-\frac{1}{2}\right)^{n-1}\]This formula helps us calculate any term needed in the sequence using straightforward multiplication and exponentiation.

Nth Term Calculation

Finding a specific term in a geometric sequence is easy once you have the general term formula. Let’s find the fifth term \(a_5\) from our sequence using this formula. The steps are simple:

• Start with the general term formula: \(a_n = 10 \cdot \left(-\frac{1}{2}\right)^{n-1}\).
• Set \(n = 5\) to find the fifth term.
• Substitute these values: \(a_5 = 10 \cdot \left( -\frac{1}{2} \right)^{5-1} = 10 \cdot \left( -\frac{1}{2} \right)^4\).

Now carry out the calculation:- Compute \(\left( -\frac{1}{2} \right)^4\) which simplifies to \(\frac{1}{16}\), since the exponent is even, the result is positive.- Multiply \(10\) by \(\frac{1}{16}\) to get:\[a_5 = \frac{10}{16} = \frac{5}{8}\]This expression shows how precise calculations using the general term formula easily allow us to determine any term in a geometric sequence.