Arithmetic Sequences General Term, Common Difference, 7th Term, Mean, And Sum
Arithmetic sequences, a fundamental concept in mathematics, are characterized by a constant difference between consecutive terms. This article provides a detailed exploration of five distinct arithmetic sequences, guiding you through the process of identifying the general term, common difference, 7th term, mean, and sum of each sequence.
1. Deciphering the Arithmetic Sequence: 8, 13, 18, 23, 28
Let's embark on our journey by dissecting the arithmetic sequence: 8, 13, 18, 23, 28. Our mission is to unearth the underlying patterns and extract key characteristics of this sequence.
Finding the General Term
The general term, often denoted as an, is a formula that allows us to calculate any term in the sequence. To determine the general term, we first need to identify the common difference, d, which is the constant value added to each term to obtain the next term. In this sequence, the common difference is 13 - 8 = 5.
The general term of an arithmetic sequence is given by the formula:
an = a1 + (n - 1) d
where a1 is the first term and n is the term number. In our case, a1 = 8 and d = 5. Substituting these values into the formula, we get:
an = 8 + (n - 1) * 5
Simplifying the expression, we arrive at the general term:
an = 5n + 3
This formula empowers us to calculate any term in the sequence by simply plugging in the desired term number, n.
Unveiling the Common Difference
As we established earlier, the common difference, d, is the constant value added to each term to generate the subsequent term. In this sequence, the common difference is 5, as each term is 5 greater than the preceding term.
Calculating the 7th Term (A7)
To find the 7th term, a7, we can utilize the general term formula we derived earlier:
a7 = 5 * 7 + 3
Evaluating this expression, we find that a7 = 38.
Determining the Mean of the Sequence
The mean of an arithmetic sequence is the average of all the terms. For a finite arithmetic sequence, the mean can be calculated by averaging the first and last terms. In this sequence, the first term is 8 and the last term is 28. Therefore, the mean is:
Mean = (8 + 28) / 2 = 18
Computing the Sum of the Sequence
The sum of an arithmetic sequence is the total of all the terms. The sum of a finite arithmetic sequence can be calculated using the formula:
Sum = (n / 2) * (a1 + an)
where n is the number of terms, a1 is the first term, and an is the last term. In this sequence, there are 5 terms, a1 = 8, and a5 = 28. Substituting these values into the formula, we get:
Sum = (5 / 2) * (8 + 28) = 90
2. Analyzing the Arithmetic Sequence: 1, 4, 7, 10, 13
Now, let's turn our attention to the arithmetic sequence: 1, 4, 7, 10, 13. We will follow the same steps as before to determine the general term, common difference, 7th term, mean, and sum of this sequence.
Deriving the General Term
The common difference, d, in this sequence is 4 - 1 = 3. Using the general term formula, an = a1 + (n - 1) d, with a1 = 1 and d = 3, we get:
an = 1 + (n - 1) * 3
Simplifying the expression, the general term becomes:
an = 3n - 2
Identifying the Common Difference
The common difference, d, is the constant value added to each term. In this sequence, the common difference is 3.
Calculating the 7th Term (A7)
Using the general term formula, we can find the 7th term:
a7 = 3 * 7 - 2
Therefore, a7 = 19.
Determining the Mean of the Sequence
The mean of this sequence is the average of the first and last terms:
Mean = (1 + 13) / 2 = 7
Computing the Sum of the Sequence
Using the sum formula, Sum = (n / 2) * (a1 + an), with n = 5, a1 = 1, and a5 = 13, we get:
Sum = (5 / 2) * (1 + 13) = 35
3. Exploring the Arithmetic Sequence: 2, 9, 16, 23, 30
Let's continue our exploration with the arithmetic sequence: 2, 9, 16, 23, 30. We will apply the same methods to uncover its general term, common difference, 7th term, mean, and sum.
Establishing the General Term
First, we find the common difference, d, which is 9 - 2 = 7. Then, using the general term formula, an = a1 + (n - 1) d, with a1 = 2 and d = 7, we have:
an = 2 + (n - 1) * 7
Simplifying the expression, we obtain the general term:
an = 7n - 5
Pinpointing the Common Difference
The common difference, d, in this sequence is 7, as each term increases by 7.
Determining the 7th Term (A7)
Using the general term formula, we can calculate the 7th term:
a7 = 7 * 7 - 5
Thus, a7 = 44.
Finding the Mean of the Sequence
The mean of this sequence is the average of the first and last terms:
Mean = (2 + 30) / 2 = 16
Computing the Sum of the Sequence
Using the sum formula, Sum = (n / 2) * (a1 + an), with n = 5, a1 = 2, and a5 = 30, we get:
Sum = (5 / 2) * (2 + 30) = 80
4. Investigating the Arithmetic Sequence: 9, 15, 21, 27, 33
Now, let's delve into the arithmetic sequence: 9, 15, 21, 27, 33. We will follow the established procedure to determine its general term, common difference, 7th term, mean, and sum.
Defining the General Term
The common difference, d, in this sequence is 15 - 9 = 6. Using the general term formula, an = a1 + (n - 1) d, with a1 = 9 and d = 6, we get:
an = 9 + (n - 1) * 6
Simplifying the expression, the general term is:
an = 6n + 3
Identifying the Common Difference
The common difference, d, is the constant increment between terms, which is 6 in this sequence.
Computing the 7th Term (A7)
Using the general term formula, we can calculate the 7th term:
a7 = 6 * 7 + 3
Therefore, a7 = 45.
Determining the Mean of the Sequence
The mean of this sequence is the average of the first and last terms:
Mean = (9 + 33) / 2 = 21
Calculating the Sum of the Sequence
Using the sum formula, Sum = (n / 2) * (a1 + an), with n = 5, a1 = 9, and a5 = 33, we get:
Sum = (5 / 2) * (9 + 33) = 105
5. Dissecting the Arithmetic Sequence: -1, 4, 9
Finally, let's examine the arithmetic sequence: -1, 4, 9. We will apply our knowledge to determine its general term, common difference, 7th term, mean, and sum.
Formulating the General Term
The common difference, d, in this sequence is 4 - (-1) = 5. Applying the general term formula, an = a1 + (n - 1) d, with a1 = -1 and d = 5, we have:
an = -1 + (n - 1) * 5
Simplifying the expression, the general term is:
an = 5n - 6
Recognizing the Common Difference
The common difference, d, is the constant value added to each term, which is 5 in this sequence.
Determining the 7th Term (A7)
Using the general term formula, we can find the 7th term:
a7 = 5 * 7 - 6
Hence, a7 = 29.
Calculating the Mean of the Sequence
Since this sequence has only three terms, the mean is calculated by summing all the terms and dividing by 3:
Mean = (-1 + 4 + 9) / 3 = 4
Computing the Sum of the Sequence
The sum of this sequence is simply the sum of the three terms:
Sum = -1 + 4 + 9 = 12
Conclusion
Through this comprehensive analysis of five distinct arithmetic sequences, we have demonstrated the methods for determining the general term, common difference, 7th term, mean, and sum. These fundamental concepts provide a solid foundation for understanding and working with arithmetic sequences in various mathematical contexts.
