Ex 4: Find The Common Difference If $a_{10} - A_{20} = 70$.
Introduction
In an arithmetic sequence, each term is obtained by adding a fixed constant to the previous term. This constant is known as the common difference. In this problem, we are given the difference between the 10th and 20th terms of an arithmetic sequence, and we need to find the common difference.
Understanding Arithmetic Sequences
An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. The general form of an arithmetic sequence is:
a, a + d, a + 2d, a + 3d, ...
where 'a' is the first term and 'd' is the common difference.
The Formula for the nth Term of an Arithmetic Sequence
The formula for the nth term of an arithmetic sequence is:
an = a + (n - 1)d
where 'an' is the nth term, 'a' is the first term, 'n' is the term number, and 'd' is the common difference.
Given Information
We are given that the difference between the 10th and 20th terms of an arithmetic sequence is 70. Mathematically, this can be expressed as:
a10 - a20 = 70
Using the Formula to Find the Common Difference
We can use the formula for the nth term of an arithmetic sequence to find the common difference. We know that:
a10 = a + (10 - 1)d a20 = a + (20 - 1)d
Substituting these expressions into the equation a10 - a20 = 70, we get:
(a + 9d) - (a + 19d) = 70
Simplifying the equation, we get:
-10d = 70
Dividing both sides by -10, we get:
d = -7
Conclusion
Therefore, the common difference of the arithmetic sequence is -7.
Example Use Case
Suppose we want to find the 15th term of the arithmetic sequence. We can use the formula for the nth term of an arithmetic sequence:
a15 = a + (15 - 1)d a15 = a + 14d
Substituting the value of 'd' we found earlier, we get:
a15 = a + 14(-7) a15 = a - 98
Therefore, the 15th term of the arithmetic sequence is a - 98.
Step-by-Step Solution
- Write down the equation a10 - a20 = 70.
- Use the formula for the nth term of an arithmetic sequence to express a10 and a20 in terms of 'a' and 'd'.
- Substitute these expressions into the equation a10 - a20 = 70.
- Simplify the equation to get -10d = 70.
- Divide both sides by -10 to get d = -7.
Key Takeaways
- The common difference of an arithmetic sequence is a fixed constant that is added to each term to get the next term.
- The formula for the nth term of an arithmetic sequence is an = a + (n - 1)d.
- We can use the formula to find the common difference by substituting the expressions for a10 and a20 into the equation a10 - a20 = 70.
Frequently Asked Questions
- What is the common difference of an arithmetic sequence?
- The common difference is a fixed constant that is added to each term to get the next term.
- How do I find the common difference of an arithmetic sequence?
- You can use the formula for the nth term of an arithmetic sequence to find the common difference by substituting the expressions for a10 and a20 into the equation a10 - a20 = 70.
- What is the formula for the nth term of an arithmetic sequence?
- The formula for the nth term of an arithmetic sequence is an = a + (n - 1)d.
Q&A: Arithmetic Sequences and Series =====================================
- The formula for the nth term of an arithmetic sequence is an = a + (n - 1)d.
Introduction
Arithmetic sequences and series are fundamental concepts in mathematics, and understanding them is crucial for solving problems in various fields, including finance, engineering, and science. In this article, we will answer some frequently asked questions about arithmetic sequences and series.
Q: What is an arithmetic sequence?
A: An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. The general form of an arithmetic sequence is:
a, a + d, a + 2d, a + 3d, ...
where 'a' is the first term and 'd' is the common difference.
Q: What is the common difference in an arithmetic sequence?
A: The common difference is a fixed constant that is added to each term to get the next term. It is denoted by 'd' and is the same between any two consecutive terms.
Q: How do I find the nth term of an arithmetic sequence?
A: You can use the formula for the nth term of an arithmetic sequence, which is:
an = a + (n - 1)d
where 'an' is the nth term, 'a' is the first term, 'n' is the term number, and 'd' is the common difference.
Q: What is the formula for the sum of an arithmetic series?
A: The formula for the sum of an arithmetic series is:
Sn = n/2 (a + l)
where 'Sn' is the sum of the first 'n' terms, 'a' is the first term, 'l' is the last term, and 'n' is the number of terms.
Q: How do I find the sum of an arithmetic series?
A: You can use the formula for the sum of an arithmetic series, which is:
Sn = n/2 (a + l)
where 'Sn' is the sum of the first 'n' terms, 'a' is the first term, 'l' is the last term, and 'n' is the number of terms.
Q: What is the formula for the sum of an infinite arithmetic series?
A: The formula for the sum of an infinite arithmetic series is:
S = a / (1 - r)
where 'S' is the sum of the infinite series, 'a' is the first term, and 'r' is the common ratio.
Q: How do I find the sum of an infinite arithmetic series?
A: You can use the formula for the sum of an infinite arithmetic series, which is:
S = a / (1 - r)
where 'S' is the sum of the infinite series, 'a' is the first term, and 'r' is the common ratio.
Q: What is the formula for the sum of a finite arithmetic series?
A: The formula for the sum of a finite arithmetic series is:
Sn = n/2 (a + l)
where 'Sn' is the sum of the first 'n' terms, 'a' is the first term, 'l' is the last term, and 'n' is the number of terms.
Q: How do I find the sum of a finite arithmetic series?
A: You can use the formula for the sum of a finite arithmetic series, which is:
Sn = n/2 (a + l)
where 'Sn' is the sum of the first 'n' terms, 'a' is the first term, 'l' is the last term, and 'n' is the number of terms.
Q: What is the formula for the nth term of an arithmetic sequence in terms of the sum of the first 'n' terms?
A: The formula for the nth term of an arithmetic sequence in terms of the sum of the first 'n' terms is:
an = Sn - (n - 1)d
where 'an' is the nth term, 'Sn' is the sum of the first 'n' terms, 'd' is the common difference, and 'n' is the term number.
Q: How do I find the nth term of an arithmetic sequence in terms of the sum of the first 'n' terms?
A: You can use the formula for the nth term of an arithmetic sequence in terms of the sum of the first 'n' terms, which is:
an = Sn - (n - 1)d
where 'an' is the nth term, 'Sn' is the sum of the first 'n' terms, 'd' is the common difference, and 'n' is the term number.
Conclusion
Arithmetic sequences and series are fundamental concepts in mathematics, and understanding them is crucial for solving problems in various fields. In this article, we have answered some frequently asked questions about arithmetic sequences and series, including the formula for the nth term, the sum of an arithmetic series, and the sum of an infinite arithmetic series.