Problem 5 - Precalculus Mathematics for Calculus

Question

The \(n\) th term of a sequence is given. (a) Find the first five terms of the sequence. (b) What is the common ratio \(r ?\) (c) Graph the terms you found in (a). $$a_{n}=5(2)^{n-1}$$

Step-by-Step Solution

Answer

The first five terms are 5, 10, 20, 40, 80. The common ratio is 2.

Step 1: Substitute to Find the First Term

To find the first term, substitute \( n = 1 \) into the formula. \[ a_1 = 5(2)^{1-1} = 5(2)^0 = 5 \times 1 = 5 \] So, the first term is 5.

Step 2: Substitute to Find the Second Term

To find the second term, substitute \( n = 2 \) into the formula. \[ a_2 = 5(2)^{2-1} = 5(2)^1 = 5 \times 2 = 10 \] Thus, the second term is 10.

Step 3: Substitute to Find the Third Term

To find the third term, substitute \( n = 3 \) into the formula. \[ a_3 = 5(2)^{3-1} = 5(2)^2 = 5 \times 4 = 20 \] Therefore, the third term is 20.

Step 4: Substitute to Find the Fourth Term

To find the fourth term, substitute \( n = 4 \) into the formula. \[ a_4 = 5(2)^{4-1} = 5(2)^3 = 5 \times 8 = 40 \] Hence, the fourth term is 40.

Step 5: Substitute to Find the Fifth Term

To find the fifth term, substitute \( n = 5 \) into the formula. \[ a_5 = 5(2)^{5-1} = 5(2)^4 = 5 \times 16 = 80 \] Therefore, the fifth term is 80.

Step 6: Identify the Common Ratio

The common ratio \( r \) is the factor by which we multiply one term to get the next. In this sequence, we multiply by 2 each time, so \( r = 2 \).

Step 7: Graph the Terms

To graph the sequence, plot the first five terms on a graph with the horizontal axis representing the term number \( n \) and the vertical axis representing the term value \( a_n \). Plot the points: \( (1, 5) \), \( (2, 10) \), \( (3, 20) \), \( (4, 40) \), \( (5, 80) \). Connect these points to show the exponential growth pattern.

Key Concepts

Geometric Sequence

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the "common ratio." This can be summarized with the formula for the nth term as \( a_n = a_1 \cdot r^{n-1} \), where:

• \( a_1 \) is the first term of the sequence,
• \( r \) is the common ratio,
• \( n \) is the term number.

In the exercise, the sequence was given as \( a_n = 5(2)^{n-1} \). Here, 5 is the first term \( a_1 \) and 2 is the common ratio \( r \). To better understand this sequence, let's break down how the terms are calculated:

• The first term, \( a_1 = 5 \).
• The second term, \( a_2 = 10 \), is obtained by multiplying the first term by 2.
• The third term, \( a_3 = 20 \), results from multiplying the second term by 2 again, and so forth.

By continuing this pattern, you can generate any term in the sequence.

Common Ratio

The common ratio in a geometric sequence is pivotal to understanding how the sequence progresses. It's the consistent factor that you multiply each term by to get to the next term. In the expression \( a_{n}=5(2)^{n-1} \), you are multiplying each term by 2, which is the common ratio \( r \).
To verify that 2 is indeed the common ratio:

• Calculate \( \frac{a_2}{a_1} \): \( \frac{10}{5} = 2 \).
• Check \( \frac{a_3}{a_2} \): \( \frac{20}{10} = 2 \).
• Continue with \( \frac{a_4}{a_3} \): \( \frac{40}{20} = 2 \), and also \( \frac{a_5}{a_4} \): \( \frac{80}{40} = 2 \).

Each division shows that the next term is twice the previous one, confirming that the common ratio \( r \) is 2. Understanding the common ratio helps in predicting future terms of the sequence without fully calculating each one.

Graphing Sequences

Graphing sequences can visually demonstrate how they behave over time. For a geometric sequence, the graph reveals an exponential growth pattern if the common ratio is greater than 1.
To graph the terms of the sequence \( a_{n}=5(2)^{n-1} \), you'll plot each term on a coordinate plane. The horizontal axis will represent the term number \( n \), while the vertical axis will show the value of \( a_n \).Here’s how the points align:

• The point corresponding to \( n = 1 \) is \( (1, 5) \).
• At \( n = 2 \), the point \( (2, 10) \) is plotted.
• \( (3, 20) \) appears at \( n = 3 \).
• Continue with \( (4, 40) \) and \( (5, 80) \).

Once plotted, connect the points. You'll notice that they form a curve, illustrating how each term is exponentially growing by the common ratio. This pattern is a key characteristic of geometric sequences and helps in understanding the rapid increase in values.