Problem 59 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write the first five terms of the sequence defined recursively. Use the pattern to write the \(n\) th term of the sequence as a function of \(n .\) (Assume \(n\) begins with 1.) $$a_{1}=6, a_{k+1}=a_{k}+2$$

Step-by-Step Solution

Answer

The first five terms are: 6, 8, 10, 12, 14. The nth term is \(a_n = n + 5\).

Step 1: Generate Sequence Terms

First, apply the recursive formula to generate the first five terms of the sequence. Given \(a_1=6\) and \(a_{k+1} = a_k + 2\), this means for every term after the first, add 2 to the previous term. Applying this gives: \(a_1 = 6\), \(a_2 = a_1 + 2 = 8\), \(a_3 = a_2 + 2 = 10\), \(a_4 = a_3 + 2 = 12\), \(a_5 = a_4 + 2 = 14\).

Step 2: Deduce the nth term

Next, observe the pattern in the sequence. Notice that the difference between the term number and term value is constant (the value of a term is always 4 more than its number). Therefore, the nth term of the sequence, \(a_n\), can be obtained by adding 4 to the term number, which is expressed as \(a_n = n + 4\).

Key Concepts

Arithmetic Sequences

An arithmetic sequence is a sequence of numbers where each term after the first is found by adding a fixed, constant value to the previous term. This constant value is called the "common difference." For example, in the sequence provided, the first term is 6 and the rule to find the next term is to add 2 to the previous one. Hence, the sequence goes 6, 8, 10, 12, and so on.

Here, the common difference is 2. This means the sequence climbs at a steady, predictable pace by adding the same amount each time.

• Initial term (\(a_1\)) is 6.
• Common difference (\(d\)) is 2, meaning the same number is added to each preceding term.

Understanding arithmetic sequences is essential as they lay the groundwork for more complex mathematical concepts and help in recognizing patterns in numbers.

nth-term Formula

The nth-term formula, also known as the general term, defines the position of any term in a sequence directly rather than relying on previous terms. This formula allows you to calculate the value of any term without having to list all preceding terms.

For arithmetic sequences, the nth-term formula can be written as:

• \( a_n = a_1 + (n-1) \, d \)

Where \( a_n \) is the nth term, \( a_1 \) the first term, \( n \) the position of the term, and \( d \) the common difference. In our sequence, where \( a_1 = 6\) and \( d = 2\), the nth-term formula can be simplified to: \( a_n = 6 + (n-1) \, 2\), which simplifies further to: \( a_n = 4 + 2n\).
This compact representation helps in predicting any term's value instantly, greatly simplifying calculations in larger series.

Sequence Pattern Identification

Spotting patterns in sequences is a powerful tool for determining how a sequence progresses. In the given exercise, recognizing that each subsequent term increases by a fixed amount (common difference) flagged it as an arithmetic sequence.

This recognition forms the basis for easier calculation and understanding.

• Observe the change between terms (constant difference here is 2).
• Recognize the formulaic pattern (in the exercise, outcome is simplified to \( a_n = n+4 \)).
• Identify the relationship between the position (n) and its term value.

When you understand the pattern, you can establish simple expressions and use shortcuts like the general formula to access any term rapidly. Developing this skill is crucial as it allows for both quick assessments and deeper deductions in math and beyond.