Problem 31 - Algebra 2 and Trigonometry

Question

In \(31-39,\) write the first five terms of each sequence. $$ a_{1}=5, a_{n}=a_{n-1}+1 $$

Step-by-Step Solution

Answer

The first five terms are 5, 6, 7, 8, and 9.

Step 1: Understanding the Sequence Rule

The exercise specifies a sequence where the first term is given as \( a_1 = 5 \) and each subsequent term is found by adding 1 to the previous term. Mathematically, this is expressed as \( a_n = a_{n-1} + 1 \).

Step 2: Calculate Second Term

Use the rule \( a_2 = a_1 + 1 \). Substitute \( a_1 = 5 \) into the equation: \( a_2 = 5 + 1 = 6 \). The second term is 6.

Step 3: Calculate Third Term

Next, calculate the third term using the rule \( a_3 = a_2 + 1 \). Substitute \( a_2 = 6 \) into the equation: \( a_3 = 6 + 1 = 7 \). The third term is 7.

Step 4: Calculate Fourth Term

Use the sequence's rule for the fourth term, \( a_4 = a_3 + 1 \). Substitute \( a_3 = 7 \): \( a_4 = 7 + 1 = 8 \). The fourth term is 8.

Step 5: Calculate Fifth Term

Finally, apply the rule to find the fifth term, \( a_5 = a_4 + 1 \). Substitute \( a_4 = 8 \): \( a_5 = 8 + 1 = 9 \). The fifth term is 9.

Key Concepts

Sequence Rule

In an arithmetic sequence, the Sequence Rule is vital as it defines how each term in the sequence is generated. This rule is typically represented with the formula:

• For the first term, we denote it as \( a_1 \).
• The nth term, \( a_n \), depends directly on its predecessor, \( a_{n-1} \).
• Typically, this involves adding a constant "difference" to the previous term.

In our given exercise, the sequence is determined by the simplified formula \( a_n = a_{n-1} + 1 \). Here, the constant difference between terms is 1. This pattern continues infinitely, and each term grows incrementally by this fixed value.The purpose of understanding the sequence rule is to seamlessly calculate subsequent terms without the need to individually solve for each one — you simply apply the rule on any known term to find the next.

Algebra 2

Within the context of Algebra 2, sequences, especially arithmetic sequences, are a foundational topic. They demonstrate how algebraic thinking can simplify this iterative process.
In this exercise, our goal is to express a recurring relationship between terms using algebraic formulas. Here:

• We started with the initial term \( a_1 = 5 \).
• Every term is derived from adding 1 to the previous one, showcasing the principle of progression using algebra.

Algebra 2 concepts teach us how to generalize mathematical problems and provide systematic strategies like this sequence rule. This encourages analytical skills by fostering abilities to manipulate expressions and solve equations. Understanding how to work with sequences not only aids in solving direct problems but lays the groundwork for more advanced topics in algebra.

Arithmetic Progression

An arithmetic progression (AP) is a type of sequence where the difference between any two successive terms is constant, known as the common difference.

• In our exercise, the common difference is 1, confirming it is an arithmetic progression.
• The first term \( a_1 \) starts the sequence, and each subsequent term is computed by adding the common difference to the previous term.

To write the first five terms of the sequence in this problem, we utilized the common difference of 1:

• The sequence begins at 5: \( a_1 = 5 \).
• Adding 1 successively, we find: \( a_2 = 6, a_3 = 7, a_4 = 8, \) and \( a_5 = 9 \).

Understanding how to identify and continue an arithmetic progression is essential because it forms the basis for more complex mathematical concepts and is widely applicable in various real-world scenarios, such as calculating interest rates in finance or anticipating schedules.