Identifying the Next Term in the Sequence -4, -1, 2, 5, 8

In this article, we will explore the pattern of the sequence -4, -1, 2, 5, 8… and determine the next term. This exercise will involve understanding arithmetic progressions and calculating the common difference.

Sequence Analysis

The given sequence is -4, -1, 2, 5, 8…. To identify the next term, we need to determine the rule governing the sequence. In this case, we observe that each term is 3 more than the previous term.

Step-by-Step Solution

Let's break down the pattern:

First Term: -4 Second Term: -1 Third Term: 2 Fourth Term: 5 Fifth Term: 8

We notice that each term is obtained by adding 3 to the previous term. This is a characteristic of an arithmetic progression, where each term is the sum of the previous term and a constant difference, denoted as 'd'. Here, the common difference 'd' is 3.

Using the Formula for Arithmetic Progression

The general formula for the nth term of an arithmetic progression is given by:

[ a_n a_1 (n-1) times d ]

Where:

( a_n ) is the nth term ( a_1 ) is the first term ( d ) is the common difference ( n ) is the term number

Given the first term ( a_1 -4 ) and the common difference ( d 3 ), we can find the sixth term (( a_6 )) by substituting ( n 6 ).

Solution Calculation

Let's calculate the sixth term:

begin{align*} a_6 -4 (6-1) times 3 [0.5em] -4 5 times 3 [0.5em] -4 15 [0.5em] 11 end{align*}

Following this pattern, the next term in the sequence is 11. Therefore, the next term in the series -4, -1, 2, 5, 8 is 11.

Verification Using Different Methods

To further verify, we can use a brute force approach or any iterative method. Here’s a simple verification using the J programming language:

J Programming Example

Using the J programming language:

_4 3 i.8

The output will be:

_4 _1 2 5 8 11 14 17

This confirms that the next term is indeed 11.

Conclusion

The next term in the sequence -4, -1, 2, 5, 8 is 11. This sequence is an arithmetic progression with a common difference of 3. By understanding and applying the formula for arithmetic progressions, we can easily determine any term in the sequence without the need for complicated calculations.