Problem 5 - 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? $$a=3, \quad r=5$$

Step-by-Step Solution

Answer

The fourth term is 375.

Step 1: Recall the Formula for the Geometric Sequence

The formula for the \(n\) th term of a geometric sequence is given by \( a_n = a \cdot r^{n-1} \), where \(a\) is the first term, and \(r\) is the common ratio.

Step 2: Identify the Values

We're given the first term \(a = 3\) and the common ratio \(r = 5\). We need to find the fourth term, so we'll use \(n = 4\).

Step 3: Substitute into the Formula

Substitute \(a = 3\), \(r = 5\), and \(n = 4\) into the formula: \( a_4 = 3 \cdot 5^{4-1} \).

Step 4: Calculate the Exponent

Calculate \(5^{4-1} = 5^3 = 125\).

Step 5: Multiply to Find the Fourth Term

Now, multiply by the first term: \(a_4 = 3 \cdot 125 = 375\). Therefore, the fourth term is 375.

Key Concepts

nth term

In a geometric sequence, the "nth term" represents the position of a specific element when counting from the beginning of the sequence. The formula to find the nth term is:

• \( a_n = a \cdot r^{n-1} \)

Here, \(a\) is the first term of the sequence, \(r\) is the common ratio, and \(n\) is the term position number.
This formula is fundamental because it allows us to calculate any term in the sequence without listing all preceding terms.
In our exercise, calculating the fourth term means substituting \(a = 3\), \(r = 5\), and \(n = 4\) into the formula. This guides us naturally to the final result, illustrating how each component interacts to expand the sequence linearly based on exponential growth.

first term

The "first term" of a geometric sequence, denoted as \(a\), is the starting point of the sequence. It acts as the sequence's foundation.
Every other term in the sequence is derived from this initial value by repeated multiplication with the common ratio.

• In our example, the first term \(a\) is given as 3.

This establishes the sequence's scale and direction from the outset. Knowing the first term is crucial because it defines the baseline from which the entire progression is built. Without it, we would not have a reference point to apply the ratio.

common ratio

The "common ratio" in a geometric sequence is what you multiply each term by to get to the next term. It is denoted as \(r\). Each term after the first is generated by multiplying the previous term by this common ratio.
This property creates the sequence's consistent pattern of growth or decay.

• For our problem, the common ratio is given as 5.

It is essential to maintain the uniformity of multiplication throughout the sequence to ensure it forms a proper geometric sequence.
Understanding the common ratio allows us to easily predict future terms and adapt the process depending on the size and direction the sequence must take according to our beginning term.