Problem 11 - Precalculus Mathematics for Calculus

Question

Find the \(n\) th term of the geometric sequence with given first term \(a\) and common ratio \(r .\) What is the fourth term? $$a=\frac{5}{2}, \quad r=-\frac{1}{2}$$

Step-by-Step Solution

Answer

The fourth term is \(-\frac{5}{16}\).

Step 1: Understanding the General Formula

Recall that the formula for the \(n\)th term of a geometric sequence is given by \(a_n = a \, r^{n-1}\). Here, \(a\) is the first term, \(r\) is the common ratio, and \(n\) is the term number we want to find.

Step 2: Identifying Given Values

From the problem, we have \(a = \frac{5}{2}\) and \(r = -\frac{1}{2}\). We need to find the fourth term in the sequence, meaning \(n = 4\).

Step 3: Substituting Values into the Formula

Substitute the given values into the formula: \(a_n = a \, r^{n-1}\). We have \(a_4 = \frac{5}{2} \left(-\frac{1}{2}\right)^{3}\).

Step 4: Calculating the Common Ratio Raised to Power

Calculate \(\left(-\frac{1}{2}\right)^{3}\). This is \(-\frac{1}{8}\) because \((-1)^3 = -1\) and \(\left(\frac{1}{2}\right)^3 = \frac{1}{8}\).

Step 5: Finding the Fourth Term

Multiply by the first term: \(a_4 = \frac{5}{2} \times -\frac{1}{8}\). This equals \(-\frac{5}{16}\).

Key Concepts

nth term formula

In a geometric sequence, each term after the first is found by multiplying the previous one by a fixed, non-zero number known as the **common ratio**. To find any term in this sequence, we use the formula for the nth term: \(a_n = a \cdot r^{n-1}\). Here, \(a\) represents the first term, \(r\) is the common ratio, and \(n\) is the term number that we are trying to find.

This formula is essential because it provides a direct way to access any term in the sequence without needing to write out all previous terms. Simply plug in the values you have for the first term, common ratio, and the specific term number you are seeking to calculate that term directly.

For our original exercise, since we know the first term \(a = \frac{5}{2}\) and the common ratio \(r = -\frac{1}{2}\), we can substitute these values into the formula to find any desired term.

common ratio

The **common ratio** of a geometric sequence is a crucial element because it defines the pattern of the sequence's growth or decay. In simpler terms, it is the constant factor by which each term of the sequence is multiplied to get the next term.

Determining the common ratio is straightforward when you have consecutive terms in the sequence: divide any term by the previous one. The result is the common ratio. A positive common ratio means the terms consistently increase or decrease by that multiple, while a negative common ratio leads to an alternation of term signs in the sequence.

For the exercise at hand, we have a common ratio \(r = -\frac{1}{2}\). This implies that each term is half of the preceding term and alternately negative, providing a distinctive alternating pattern as the sequence progresses.

fourth term calculation

Calculating specific terms, like the **fourth term**, in a geometric sequence uses the nth term formula. To solve for \(a_4\), we follow these steps:

• Substitute the given values into the nth term formula: \(a_4 = \frac{5}{2} \cdot \left( -\frac{1}{2} \right)^{3}\).
• The power \(3\) reflects \(n-1\), as we are seeking the fourth term.

Let's handle the exponent first. The calculation \(\left(-\frac{1}{2}\right)^3\) results in \(-\frac{1}{8}\) because \((-1)^3 = -1\) and \(\left(\frac{1}{2}\right)^3 = \frac{1}{8}\).

Finally, multiplying the first term by this result gives \(a_4 = \frac{5}{2} \times -\frac{1}{8} = -\frac{5}{16}\). Therefore, the fourth term is \(-\frac{5}{16}\), reflecting both the effect of the common ratio and the power sequence.