Problem 5 - Algebra and Trigonometry

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 is 2.

Step 1: Substitute Values for n

To find the first five terms of the sequence, we substitute values starting from \( n = 1 \) to \( n = 5 \) into the given formula \( a_n = 5 + 2(n-1) \).

Step 2: Calculate the First Term

Substitute \( n = 1 \) into \( a_n = 5 + 2(n-1) \), giving us \( a_1 = 5 + 2(1-1) = 5 \).

Step 3: Calculate the Second Term

Substitute \( n = 2 \) into the sequence formula, giving us \( a_2 = 5 + 2(2-1) = 7 \).

Step 4: Calculate the Third Term

Substitute \( n = 3 \) into the sequence formula, giving us \( a_3 = 5 + 2(3-1) = 9 \).

Step 5: Calculate the Fourth Term

Substitute \( n = 4 \) into the sequence formula, yielding \( a_4 = 5 + 2(4-1) = 11 \).

Step 6: Calculate the Fifth Term

Substitute \( n = 5 \) into the sequence formula, yielding \( a_5 = 5 + 2(5-1) = 13 \).

Step 7: Identify the Sequence Terms

The first five terms of the sequence are \( 5, 7, 9, 11, 13 \).

Step 8: Determine the Common Difference

The common difference \( d \) is the difference between consecutive terms. Here, \( d = 7 - 5 = 2 \).

Step 9: Graph the Sequence Terms

Plot the terms found in Step 7 on a graph, with the x-axis representing the term (1, 2, 3, 4, 5) and the y-axis representing the sequence values (5, 7, 9, 11, 13).

Key Concepts

Common Difference in Arithmetic Sequences

In an arithmetic sequence, the common difference is the amount by which each term increases (or decreases) from the previous term. This difference is constant throughout the sequence.To determine the common difference:

• Choose any two consecutive terms from the sequence.
• Subtract the earlier term from the later term.
• The result is the common difference \( d \).

These steps show that for the sequence with terms 5, 7, 9, 11, 13, the common difference \( d \) is 2, as calculated by \( 7 - 5 = 2 \). Understanding this concept is crucial, as the common difference helps predict future terms and discern the sequence's growth pattern.
Recognizing the common difference allows one to easily jump between terms or assess the sequence's properties without recalculating each term individually.

Sequence Graphing: Visualizing Arithmetic Sequences

Graphing an arithmetic sequence provides a visual representation of the sequence's behavior, making it easy to identify the regular intervals of change. To graph an arithmetic sequence:

• Plot the terms on a coordinate grid, where the x-axis represents the term position and the y-axis represents the sequence value.
• Place your points according to the terms and their respective values.

For the sequence 5, 7, 9, 11, and 13, plot the points (1,5), (2,7), (3,9), (4,11), and (5,13). Connect these points to form a straight line.
This line illustrates the constant growth determined by the common difference.
The graph is a straight line because every interval between the points is equidistant, highlighting the arithmetic sequence's regularity and predictability.

Finding Sequence Terms: The Role of the Formula

In arithmetic sequences, finding specific terms relies on the sequence formula, which is designed to determine the value of any term based on its position.The general formula for finding terms in an arithmetic sequence is usually given by: \[ a_n = a_1 + (n-1) \cdot d \].In our given problem, this formula is slightly adjusted: \[ a_n = 5 + 2(n-1) \].Here's how to use the formula:

• Identify the first term \( a_1 \), and the common difference \( d \).
• Substitute the desired term number \( n \) into the formula to find the term value.

For example, to find the third term, substitute \( n = 3 \) into the formula:\[ a_3 = 5 + 2(3-1) = 9 \].This sequence formula simplifies the process of finding any term within the sequence without having to calculate each previous term.
By relying on the formula, the efficiency and convenience of examining large sequences is greatly enhanced.