Problem 6 - Precalculus: Mathematical 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? $$a=-6, \quad r=3$$

Step-by-Step Solution

Answer

The fourth term is -162.

Step 1: Identify the General Formula for the nth Term

The nth term of a geometric sequence can be found using the formula \( a_n = a \cdot r^{n-1} \), where \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the term number you want to find.

Step 2: Plug in the Given Values

We are given \( a = -6 \) and \( r = 3 \). We want to find the fourth term (\( n = 4 \)). Substitute these values into the formula: \( a_4 = -6 \cdot 3^{4-1} \).

Step 3: Simplify the Exponentiation

Calculate \( 3^{4-1} \). This is \( 3^3 \). Perform the multiplication: \( 3 \times 3 \times 3 = 27 \).

Step 4: Compute the Fourth Term

Now multiply the result from the exponentiation by the first term: \( a_4 = -6 \cdot 27 \). Calculate this multiplication: \( -6 \times 27 = -162 \).

Key Concepts

nth term formula

In geometric sequences, finding the nth term is crucial for understanding how these sequences progress visually. The core of this lies in the nth term formula: \[ a_n = a \cdot r^{n-1} \] Here, \( a_n \) represents the term you’re trying to find, \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the term's position in the sequence. This formula allows us to quickly determine any term in the sequence without listing out all previous terms. It's a time-saver and efficient, which is especially useful when dealing with large values of \( n \).
For example, when finding the fourth term given \( a = -6 \) and \( r = 3 \), the formula becomes: \[ a_4 = -6 \cdot 3^{4-1} = -6 \cdot 3^3 \] Outputting \(-162\). Utilizing this formula provides a structured approach to solve such problems consistently.

common ratio

The common ratio \( r \) is a fundamental component in understanding geometric sequences. It's the factor by which we multiply each term to get the next one. In our example, the common ratio is given as 3. This means each term is three times the preceding term.
To really grasp this concept:

• If \( a = -6 \), then the second term \( a_2 = (-6) \times 3 = -18 \).
• The third term \( a_3 = (-18) \times 3 = -54 \).
• The fourth term \( a_4 = (-54) \times 3 = -162 \).

This step-by-step multiplication shows the role of the common ratio in advancing through a geometric sequence. The constant nature of multiplication by \( r \) is what makes the sequence geometric.

first term calculation

Calculating the first term \( a \) is straightforward but essential as it sets the base for the entire sequence. In our example, the first term is \(-6\). This value is the starting point and is used in calculating the subsequent terms with the common ratio.
In geometric sequences, once the first term is known, each subsequent term is derived by multiplying the previous term by the common ratio \( r \). Hence, \( a \) is critical:

• It anchors the sequence.
• It's the first number you'll see when listing the sequence.
• Without it, applying the nth term formula would be impossible since \( a \) is an integral part of it.

This shows that even if the sequence seems complex at first, understanding and accurately determining the first term makes everything more manageable.