Finding the Term Number in an Arithmetic Sequence
Consider an arithmetic sequence where the common difference is d = 5, and we know the value of a specific term is aₙ = 40. However, we do not know the term number n. The first term of the sequence is a = 10. Let’s find the value of n.
Problem Statement
Given an arithmetic sequence with a = 10, d = 5, and aₙ = 40, find the term number n for which the value of the term is 40.
Solution
We can use the formula for the n-th term of an arithmetic sequence:
aₙ = a + (n – 1) × d (General formula)
40 = 10 + (n – 1) × 5 (Substitute given values)
40 = 10 + 5n – 5 (Distribute the 5)
35 = 5n (Combine like terms)
n = 7 (Divide by 5 to solve for n)
So the term number for which the value of the term is 40 is n = 7.
Conclusion
In this arithmetic sequence, the 7th term has the value 40. This example demonstrates how to find the term number when the term value and common difference are known, but the term number is unknown. The step-by-step explanation helps beginners understand the process of solving the problem.
Arithmetic Sequences with Negative Common Difference
An arithmetic sequence can have a negative common difference, meaning that each term decreases by a constant amount. Let’s explore an example of such a sequence.
Example: Sequence 10, 7, 4, 1, -2
Consider the arithmetic sequence 10, 7, 4, 1, -2. In this sequence, the first term is \(a = 10\), and the common difference is \(d = -3\), as each term decreases by 3.
Derivation of the Formula
We can derive the formula for the \(n\)-th term of this sequence as follows:
\(a_n = a + (n – 1) \cdot d = 10 + (n – 1) \cdot (-3) = 10 – 3n + 3 = 13 – 3n\)
So the formula for the \(n\)-th term of this specific arithmetic sequence is \(a_n = 13 – 3n\).
Using the Formula to Find Specific Terms
Example: Using the derived formula, we can find any term in the sequence 10, 7, 4, 1, -2. For instance, the 4th term is:
\(a_4 = 13 – 3 \cdot 4 = 13 – 12 = 1\)
This demonstrates how the pattern in the sequence leads to the simplified formula, and how that formula can be used to find specific terms, even when the common difference is negative.
Applications
Arithmetic sequences with negative common differences can model situations of constant decline, such as depreciation of assets, cooling of a hot object, or decreasing sales. Understanding these sequences is essential in various real-world applications.
Finding the Term Number in an Arithmetic Sequence
Consider an arithmetic sequence where the common difference is d = 5, and we know the value of a specific term is aₙ = 40. However, we do not know the term number n. The first term of the sequence is a = 10. Let’s find the value of n.
Problem Statement
Given an arithmetic sequence with a = 10, d = 5, and aₙ = 40, find the term number n for which the value of the term is 40.
Solution
We can use the formula for the n-th term of an arithmetic sequence:
aₙ = a + (n – 1) × d
Substituting the given values, we have:
40 = 10 + (n – 1) × 5
40 = 10 + 5n – 5
35 = 5n
n = 7
So the term number for which the value of the term is 40 is n = 7.
Conclusion
In this arithmetic sequence, the 7th term has the value 40. This example demonstrates how to find the term number when the term value and common difference are known, but the term number is unknown.
Deriving the Formula for the n-th Term of an Arithmetic Sequence
An arithmetic sequence is a sequence of numbers where the difference between consecutive terms is constant. This difference is called the common difference, denoted by d.
First Few Terms
Let’s write down the first few terms of an arithmetic sequence:
- 1st term: a (the first term itself)
- 2nd term: a + d (one common difference added)
- 3rd term: a + 2d (two common differences added)
- 4th term: a + 3d (three common differences added)
- And so on…
Generalizing the Pattern
We can see a pattern emerging. The n-th term is the first term plus the common difference multiplied by one less than the term number. This “one less” comes from the fact that we start with the first term itself (a) and then add the common difference (d) (n-1) times.
Formula for the n-th Term
By generalizing this pattern, we can write the formula for the n-th term as:
aₙ = a + (n – 1) × d
Conclusion
This formula allows us to find any term in the arithmetic sequence if we know the first term (a), the common difference (d), and the term number (n). The (n-1) in the formula represents the fact that we add the common difference (n-1) times to the first term to reach the n-th term.