Finding the 13th Term of the Arithmetic Sequence

Oct 13, 2025 by recoveryshake.com 96 views

Understanding Arithmetic Sequences

Arithmetic sequences are a series of numbers in which the difference between any two consecutive terms is constant. This constant difference is known as the common difference. The sequence given in the problem is 11, 13, 15, 17, _, and we are asked to find the 13th term.

Finding the 13th Term of the Arithmetic Sequence

To find the 13th term of the sequence, we need to determine the common difference between the terms. Looking at the given sequence, we can see that each term increases by 2. Therefore, the common difference is 2.

Now that we know the common difference, we can use the formula for the nth term of an arithmetic sequence:

an = a1 + (n - 1)d

where an is the nth term, a1 is the first term, n is the term number, and d is the common difference.

In this case, a1 = 11, n = 13, and d = 2. Plugging these values into the formula, we get:

a13 = 11 + (13 - 1)2 a13 = 11 + 24 a13 = 35

Therefore, the 13th term of the sequence is 35.

Understanding Arithmetic Mean

The arithmetic mean of a set of numbers is the sum of the numbers divided by the number of terms. In the second part of the problem, we are given that the arithmetic mean of 5 and c is 11.

Finding the Value of c

To find the value of c, we can use the formula for the arithmetic mean:

Arithmetic mean = (sum of terms) / (number of terms)

In this case, the arithmetic mean is 11, and the number of terms is 2. We can set up an equation using the formula:

11 = (5 + c) / 2

To solve for c, we can multiply both sides of the equation by 2:

22 = 5 + c

Next, we can subtract 5 from both sides of the equation:

17 = c

Therefore, the value of c is 17.

Real-World Applications of Arithmetic Sequences and Mean

Arithmetic sequences and mean have many real-world applications. For example, in finance, the interest on a savings account or loan is often calculated using arithmetic sequences. In statistics, the arithmetic mean is used to calculate the average of a set of numbers.

Conclusion

In conclusion, solving mathematical sequences and equations requires a strong understanding of mathematical concepts and formulas. By applying these concepts and formulas, we can solve problems such as finding the 13th term of an arithmetic sequence and determining the value of c in an equation involving the arithmetic mean.

Additional Examples and Practice Problems

Here are some additional examples and practice problems to help you practice solving arithmetic sequences and mean:

Example 1

Find the 10th term of the arithmetic sequence 2, 4, 6, 8, _, _, _, _, _, _

Solution

To find the 10th term, we need to determine the common difference between the terms. Looking at the given sequence, we can see that each term increases by 2. Therefore, the common difference is 2.

Using the formula for the nth term of an arithmetic sequence, we get:

a10 = 2 + (10 - 1)2 a10 = 2 + 18 a10 = 20

Therefore, the 10th term of the sequence is 20.

Example 2

Find the value of c in the equation:

Arithmetic mean = (5 + c) / 2

Solution

To find the value of c, we can use the formula for the arithmetic mean:

Arithmetic mean = (sum of terms) / (number of terms)

In this case, the arithmetic mean is 11, and the number of terms is 2. We can set up an equation using the formula:

11 = (5 + c) / 2

To solve for c, we can multiply both sides of the equation by 2:

22 = 5 + c

Next, we can subtract 5 from both sides of the equation:

17 = c

Therefore, the value of c is 17.

Practice Problems

Find the 15th term of the arithmetic sequence 3, 5, 7, 9, _, _, _, _, _, _, _, _, _, _
Find the value of c in the equation:

Arithmetic mean = (3 + c) / 2

Find the 20th term of the arithmetic sequence 1, 3, 5, 7, _, _, _, _, _, _, _, _, _, _, _, _, _, _, _, _

I hope these examples and practice problems help you practice solving arithmetic sequences and mean.

Frequently Asked Questions

Q: What is an arithmetic sequence?

A: An arithmetic sequence is a series of numbers in which the difference between any two consecutive terms is constant. This constant difference is known as the common difference.

Q: How do I find the common difference of an arithmetic sequence?

A: To find the common difference, look at the given sequence and find the difference between any two consecutive terms. 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