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). $$a_{n}=7(3)^{n-1}$$

Step-by-Step Solution

Answer

The first five terms are 7, 21, 63, 189, 567. The common ratio is 3.

Step 1: Understand the Formula

The formula given is \(a_n = 7(3)^{n-1}\). This formula is for a geometric sequence where \(a_1 = 7\) (the first term) and the common ratio \(r = 3\).

Step 2: Find the First Term

To find the first term, substitute \(n = 1\) into the formula: \(a_1 = 7(3)^{1-1} = 7(3)^0 = 7\).

Step 3: Find the Second Term

Substitute \(n = 2\) into the formula: \(a_2 = 7(3)^{2-1} = 7(3)^1 = 21\).

Step 4: Find the Third Term

Substitute \(n = 3\) into the formula: \(a_3 = 7(3)^{3-1} = 7(3)^2 = 63\).

Step 5: Find the Fourth Term

Substitute \(n = 4\) into the formula: \(a_4 = 7(3)^{4-1} = 7(3)^3 = 189\).

Step 6: Find the Fifth Term

Substitute \(n = 5\) into the formula: \(a_5 = 7(3)^{5-1} = 7(3)^4 = 567\).

Step 7: Determine the Common Ratio

The common ratio \(r\) is the factor by which each term is multiplied to get the next term. Here, \(r = 3\) since each term is \(3\) times the previous term.

Step 8: Graph the Sequence

Plot the values obtained from steps 2 to 6 on a coordinate plane with the x-axis representing the term number \(n\) and the y-axis representing the term value. The points to plot are \((1, 7), (2, 21), (3, 63), (4, 189), (5, 567)\). Connect these points to show the pattern.

Key Concepts

Common Ratio

In a geometric sequence, a common ratio is the factor used to multiply a term to get the next term in the sequence. For example, if you know one term and the common ratio, multiplying the term by the common ratio gives you the next term.
This concept is crucial as it defines the entire sequence. For the sequence defined by the formula \(a_n = 7(3)^{n-1}\), the common ratio is \(3\). This means each term is three times the previous one.

• Start with the first term \(a_1 = 7\).
• Next, multiply by 3 repeatedly to get the subsequent terms: \(21, 63, 189,\) and so on.

Recognizing this pattern is helpful for quickly determining any term in the sequence without calculating every previous term.

Graphing Sequences

Graphing a geometric sequence helps visualize the relationship between the term number and its value. Plotting a few terms gives a clear picture of how the sequence grows. On the graph, the x-axis represents the term number \(n\), while the y-axis shows the term's value.
For the sequence \(a_n = 7(3)^{n-1}\), the plotted points are:

• (1, 7)
• (2, 21)
• (3, 63)
• (4, 189)
• (5, 567)

When you connect these points on a graph, you notice an exponential growth pattern. The curve gets steeper as you move to higher term numbers, illustrating rapid growth typical of geometric sequences with a common ratio greater than one.

Term Calculation

Calculating terms in a geometric sequence is straightforward when using its formula. Here, we have \(a_n = 7(3)^{n-1}\). This formula means each term is calculated by multiplying the first term by the common ratio raised to an exponent. The exponent is one less than the term number \(n\).
For example:

• First term: \(7(3)^{1-1} = 7(3)^0 = 7\)
• Second term: \(7(3)^{2-1} = 21\)
• Third term: \(7(3)^{3-1} = 63\)
• And so forth.

This method enables you to find any term directly without manually calculating all the preceding terms. Once you know the common ratio and the first term, you can easily compute any term in the sequence.