Problem 36 - Algebra 2 and Trigonometry

Question

In \(31-39,\) write the first five terms of each sequence. $$ a_{1}=4, a_{n+1}=a_{n}+n $$

Step-by-Step Solution

Answer

The first five terms of the sequence are 4, 5, 7, 10, and 14.

Step 1: Identify the Sequence's Formula

We are given the first term of the sequence, \(a_1 = 4\). The recursive formula for the sequence is \(a_{n+1} = a_n + n\). This means each subsequent term is obtained by adding \(n\) to the current term \(a_n\).

Step 2: Calculate the Second Term

To find \(a_2\), we use the relation \(a_{n+1} = a_n + n\). For \(n = 1\): \[ a_2 = a_1 + 1 = 4 + 1 = 5 \]

Step 3: Calculate the Third Term

Now, using the formula with \(n = 2\), we find \(a_3\): \[ a_3 = a_2 + 2 = 5 + 2 = 7 \]

Step 4: Calculate the Fourth Term

Continuing the pattern, for \(n = 3\), calculate \(a_4\): \[ a_4 = a_3 + 3 = 7 + 3 = 10 \]

Step 5: Calculate the Fifth Term

Finally, compute \(a_5\) using \(n = 4\): \[ a_5 = a_4 + 4 = 10 + 4 = 14 \]

Step 6: Write the First Five Terms

Now that we have computed all necessary terms, the first five terms of the sequence are \(4, 5, 7, 10, 14\).

Key Concepts

Understanding Recursive Sequences

A recursive sequence is a series of numbers in which each term is defined using one or more previous terms. In this approach, you start with an initial number, known as the initial term, and apply a recursive relation to find the subsequent terms. This method is like following a recipe to achieve a delicious dish: you start with an ingredient and follow each step, adding more ingredients systematically.

Recursive sequences are inherently linked to their initial value, as this first term becomes the foundation upon which all subsequent calculations are based. The formula or rule given as part of the recursive sequence helps you determine each following term. For example, our sequence starts with the initial term, \(a_1 = 4\), and applies the recursive relation \(a_{n+1} = a_n + n\).

By adding \(n\) to the current term, you can construct the entire sequence one term at a time.

• Start from the known term (like \(4\) in our example).
• Apply the recursive formula (like adding \(n\) in our problem).
• Continue this process to obtain additional terms.

This manner of constructing sequences is powerful because it builds on itself, using each new term as the stepping stone for the next one. Understanding this concept is key to tackling problems involving sequences in mathematics.

Exploring Arithmetic Progressions

Arithmetic progression (AP) is a specific type of sequence where each term after the first is the sum of the previous term and a constant. This might sound similar to our recursive sequence, but here, each term increases by a constant difference, not by an increment that depends on the position. Though not exactly the method used in our example, understanding arithmetic progressions can be beneficial when identifying patterns in number sequences.

For an arithmetic progression, the general form of the sequence is given by:

• \(a_n = a_1 + (n-1) \cdot d\)

Here, \(d\) is the constant difference between terms. Thus, each new term is simply a direct addition of this constant to the preceding one.

Comparing this with our problem, the difference varies with the term number due to our recursive formula \(a_{n+1} = a_n + n\). Nevertheless, in both cases, logic builds step by step, taking note of a connecting rule or difference.

Role of the Initial Term

The initial term of a sequence is the starting point from which the entire sequence evolves. It's like the starting block in a race, setting the stage for everything that follows. In our case, the initial term is \(a_1 = 4\). This is crucial, as it sets the cascade of operations into motion, playing a pivotal role in the development of both recursive sequences and arithmetic progressions.

When working with a sequence, the initial term offers a concrete number from which you can apply your formula or recursive relation. For example, by using the initial value of 4 in our recursive equation, \(a_{n+1} = a_n + n\), we're able to calculate each subsequent term reliably:

• From \(a_1 = 4\), the next term becomes \(a_2 = 4 + 1 = 5\).
• Using \(a_2\), we find \(a_3 = 5 + 2 = 7\), and so on.

Without this starting value, the sequence would be indefinite and directionless. Each term needs a base to build on, and that base is lovingly provided by the initial term. It's where clarity and structure originate, ensuring the sequence is bound effectively to its defining rule.