Understanding Arithmetic Sequences and Calculating the Fifth Term

Arithmetic sequences are a key concept in algebra and mathematics, where each term is generated by adding a constant difference to the previous term. This article explores the process of identifying the common difference and calculating any term in an arithmetic sequence, using a specific example to illustrate the method.

Identifying the Common Difference

To determine if a sequence is arithmetic, we need to check if the difference between consecutive terms is constant. Given the sequence:

4x - 1, -3x - 4, -1 - 7, ...

Assuming the third term is corrected to -1 - 7, we write:

The common difference, d, is the difference between consecutive terms:

d term_2 - term_1 term_3 - term_2

Let's calculate d using the first two terms:

d (-3x - 4) - (4x - 1) -7x - 3

Now, we verify using the second and third terms:

d (-1 - 7) - (-3x - 4) -7x - 3

Since the common difference is confirmed, the sequence is indeed arithmetic.

Calculating the Fifth Term

The general formula for the nth term of an arithmetic sequence is:

tn a (n-1)d

Where:

a is the first term d is the common difference n is the term number

Given the first term a 4x - 1, the common difference d -7x - 3, and we need to find the fifth term (n 5):

t5 (4x - 1) (5 - 1)(-7x - 3)

Simplifying this expression:

t5 (4x - 1) 4(-7x - 3)

t5 4x - 1 - 28x - 12

t5 -24x - 13

Thus, the fifth term of the sequence is -24x - 13.

Understanding the Given Sequence

Let's consider the sequence as:

4x - 1, -3x - 4, -1 - 7, -17x - 10, -24x - 13

Here, the common difference d -7x - 3, and the fifth term is -24x - 13.

For clarity, another series is analyzed:

4x - 1, -3x - 4, -10 - 7, -17x - 10, -27x - 17

Here, the common difference is also -7x - 3.

The fifth term is calculated as:

4x - 1 4(-7x - 3) 4x - 1 - 28x - 12 -24x - 13

Thus, the fifth term is consistently -24x - 13.

Conclusion

Arithmetic sequences can be analyzed by identifying the common difference and using the general term formula to find any term. Whether the third term is -1 - 7 or -10 - 7, the common difference and the fifth term remain consistent.

h4Keywords:/h4ul liarithmetic sequence/li licommon difference/li literm calculation/li /ul