Problem 1 - Algebra and Trigonometry

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: Calculate the First Term

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

Step 2: Calculate the Second Term

For the second term (\(a_2\)), substitute\(n = 2\) into the formula:\[a_2 = 5(2)^{2-1} = 5(2)^1 = 5 imes 2 = 10.\]So, the second term is 10.

Step 3: Calculate the Third Term

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

Step 4: Calculate the Fourth Term

For the fourth term (\(a_4\)), substitute\(n = 4\):\[a_4 = 5(2)^{4-1} = 5(2)^3 = 5 imes 8 = 40.\]The fourth term is 40.

Step 5: Calculate the Fifth Term

For the fifth term (\(a_5\)), substitute\(n = 5\):\[a_5 = 5(2)^{5-1} = 5(2)^4 = 5 imes 16 = 80.\]The fifth term is 80.

Step 6: Determine the Common Ratio

The sequence is geometric, with each term obtained by multiplying the previous term by a common ratio from\(a_{n-1}\)to\(a_n\).Therefore, the common ratio\(r\)is given by:\[r = \frac{a_2}{a_1} = \frac{10}{5} = 2.\]The common ratio is 2.

Step 7: Graph the Sequence Terms

Graph the terms found in (a):\[\begin{align*}\text{Term}&: 1, 2, 3, 4, 5 \text{Value}&: 5, 10, 20, 40, 80 \end{align*}\]Plot these points on a graph where the x-axis represents the term number and the y-axis represents the term value.

Key Concepts

Common Ratio

In a geometric sequence, the common ratio is a crucial aspect to understand. It is the constant factor between consecutive terms. For any geometric sequence, you can calculate the common ratio by dividing a term by its preceding term. For example, in the sequence derived from the exercise, the terms are 5, 10, 20, 40, and 80. To find the common ratio, you divide 10 by 5 (the first term by the second term), which equals 2. Hence, the common ratio is 2.
This means that each term is twice the previous term. It simplifies understanding the pattern and predicting further terms in the sequence.

Term Calculation

Calculating the terms of a geometric sequence involves substituting the sequence's term number into its formula. The general formula for our given sequence is \( a_n = 5(2)^{n-1} \).
Here is how to calculate the first few terms:

• For the 1st term \( (n = 1) \): \( a_1 = 5(2)^{0} = 5 \).
• For the 2nd term \( (n = 2) \): \( a_2 = 5(2)^{1} = 10 \).
• For the 3rd term \( (n = 3) \): \( a_3 = 5(2)^{2} = 20 \).
• For the 4th term \( (n = 4) \): \( a_4 = 5(2)^{3} = 40 \).
• For the 5th term \( (n = 5) \): \( a_5 = 5(2)^{4} = 80 \).

Each of these computations follows the sequence formula, demonstrating how each subsequent term is obtained from the previous one by applying the common ratio of 2. This method can be repeated to find any further term in the sequence easily.

Graphing Sequences

Graphing a geometric sequence on a coordinate plane helps visualize the growth of the sequence. It gives a clear picture of how rapidly the sequence values increase.
To graph a sequence, follow these steps:

• Label the x-axis with the term number, and the y-axis with the value of the term.
• Plot each term from the sequence as a point on the graph. For our exercise, you plot the points (1, 5), (2, 10), (3, 20), (4, 40), and (5, 80).
• Connect the points with a line to illustrate the trend.

As you connect the points, you will notice an upward curve, characteristic of a geometric sequence, indicating exponential growth. This graph serves as a visual representation of the sequence's dynamic and highlights how each subsequent term grows as a multiple of the previous term by the common ratio.