Problem 1 - Precalculus: Mathematical for Calculus

Question

A sequence is given. (a) Find the first five terms of the sequence. (b) What is the common difference \(d ?\) (c) Graph the terms you found in (a). $$a_{n}=5+2(n-1)$$

Step-by-Step Solution

Answer

First five terms: 5, 7, 9, 11, 13. Common difference: 2.

Step 1: Identify the Formula for the Sequence

The given sequence is described by the formula \(a_{n} = 5 + 2(n-1)\). This formula will help us find each term of the sequence based on the position \(n\).

Step 2: Calculate the First Term

To find the first term \(a_1\), substitute \(n = 1\) into the formula: \[a_1 = 5 + 2(1-1) = 5 + 2 \cdot 0 = 5.\] The first term \(a_1\) is 5.

Step 3: Calculate the Second Term

To find the second term \(a_2\), substitute \(n = 2\) into the formula: \[a_2 = 5 + 2(2-1) = 5 + 2 \cdot 1 = 5 + 2 = 7.\] The second term \(a_2\) is 7.

Step 4: Calculate the Third Term

To find the third term \(a_3\), substitute \(n = 3\) into the formula: \[a_3 = 5 + 2(3-1) = 5 + 2 \cdot 2 = 5 + 4 = 9.\] The third term \(a_3\) is 9.

Step 5: Calculate the Fourth Term

To find the fourth term \(a_4\), substitute \(n = 4\) into the formula: \[a_4 = 5 + 2(4-1) = 5 + 2 \cdot 3 = 5 + 6 = 11.\] The fourth term \(a_4\) is 11.

Step 6: Calculate the Fifth Term

To find the fifth term \(a_5\), substitute \(n = 5\) into the formula: \[a_5 = 5 + 2(5-1) = 5 + 2 \cdot 4 = 5 + 8 = 13.\] The fifth term \(a_5\) is 13.

Step 7: Determine the Common Difference

The sequence is arithmetic, meaning there is a common difference between consecutive terms. Calculate it using \(a_2 - a_1\): \[d = a_2 - a_1 = 7 - 5 = 2.\] The common difference \(d\) is 2.

Step 8: Graph the Sequence Terms

On a Cartesian plane, plot the terms calculated from step 2 to step 6. Plot these points: \((1, 5), (2, 7), (3, 9), (4, 11), (5, 13)\). Each point corresponds to \(n\) and \(a_n\) with \(n\) as the x-coordinate and \(a_n\) as the y-coordinate.

Key Concepts

Common Difference

In an arithmetic sequence, the **common difference**, denoted as \(d\), is a crucial concept. It tells us how much each term increases or decreases when moving from one term to the next. This consistent interval is what defines the sequence's linear pattern.
In the given sequence, each term is generated by the formula \(a_n = 5 + 2(n-1)\). Here, the coefficient of \(n-1\) is the common difference. To determine \(d\), observe how each term grows:

• First term: \(a_1 = 5\)
• Second term: \(a_2 = 7\)
• Third term: \(a_3 = 9\)
• Fourth term: \(a_4 = 11\)
• Fifth term: \(a_5 = 13\)

Notice the constant increase by 2 between each pair of consecutive terms. Calculate it using the difference between any two consecutive terms, for example, \(a_2 - a_1 = 7 - 5 = 2\). Hence, the common difference \(d\) is 2.
This value helps predict future terms and confirms the linear progression of the sequence.

Sequence Graphing

Graphing an arithmetic sequence can visually represent the relationship between the terms and their positions. To graph the sequence, each term's index \(n\) is plotted on the x-axis, while its value \(a_n\) is plotted on the y-axis.
For our sequence with terms \((1, 5), (2, 7), (3, 9), (4, 11), (5, 13)\), mark these points on a Cartesian plane. Draw a point for each pair. This pattern reveals a straight line, indicative of a linear graph due to the constant common difference.
Once plotted, the graph showcases that the sequence progresses linearly, with a slope equal to the common difference. This visualization solidifies the understanding of how each term is calculated in an arithmetic sequence and how they consistently expand or contract.

Term Calculation

To calculate the terms in an arithmetic sequence, one uses the general formula \(a_n = a_1 + (n-1)d\). This formula helps find the \'n\'th term in the sequence by knowing:

• \(a_1\), the starting term
• \(d\), the common difference
• \(n\), the position of the term you want to find

Applying this to our sequence, notice how each term follows the pattern:
For the first term, substitute \(n=1\):
\[a_1 = 5 + 2(1-1) = 5\]

For the second term, substitute \(n=2\):
\[a_2 = 5 + 2(2-1) = 7\]

Repeat this method for subsequent terms to verify their correctness:
\[a_3 = 9, \quad a_4 = 11, \quad a_5 = 13\]
This structured approach allows for efficient calculation and verification. By using this formula, predicting any term becomes straightforward, helping students grasp the scalability of arithmetic sequences.