The Fourth Term And The Second Term Of An Arithmetic Progression Are 4 And 5, Respectively. Find:(i) The First Term And The Common Difference.(ii) The Fortieth Term Of The Progression.
Introduction
An arithmetic progression is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is called the common difference. In this article, we will discuss how to find the first term and the common difference of an arithmetic progression given the fourth term and the second term. We will also find the fortieth term of the progression.
The Formula for the nth Term of an Arithmetic Progression
The formula for the nth term of an arithmetic progression is given by:
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 fourth term (a4) is 4 and the second term (a2) is 5. We need to find the first term (a) and the common difference (d).
Step 1: Find the Common Difference
We can use the formula for the nth term to find the common difference. We know that a4 = 4 and a2 = 5. We can write two equations using the formula:
a + 3d = 4 ... (1) a + d = 5 ... (2)
Subtracting equation (2) from equation (1), we get:
2d = -1 d = -1/2
Step 2: Find the First Term
Now that we have found the common difference, we can find the first term. We can use either equation (1) or equation (2) to find the first term. Let's use equation (2):
a + d = 5 a + (-1/2) = 5 a = 5 + 1/2 a = 11/2
The First Term and the Common Difference
Therefore, the first term (a) is 11/2 and the common difference (d) is -1/2.
The Fortieth Term of the Progression
Now that we have found the first term and the common difference, we can find the fortieth term of the progression. We can use the formula for the nth term:
a40 = a + (40 - 1)d a40 = 11/2 + 39(-1/2) a40 = 11/2 - 39/2 a40 = -28/2 a40 = -14
Conclusion
In this article, we discussed how to find the first term and the common difference of an arithmetic progression given the fourth term and the second term. We also found the fortieth term of the progression. We used the formula for the nth term of an arithmetic progression to solve the problem.
Arithmetic Progression Formula
The formula for the nth term of an arithmetic progression 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.
Arithmetic Progression Example
Find the first term and the common difference of an arithmetic progression given the fourth term (a4) is 4 and the second term (a2) is 5. Find the fortieth term of the progression.
Solution
a + 3d = 4 ... (1) a + d = 5 ... (2)
Subtracting equation (2) from equation (1), we get:
2d = -1 d = -1/2
a + d = 5 a + (-1/2) = 5 a = 5 + 1/2 a = 11/2
a40 = a + (40 - 1)d a40 = 11/2 + 39(-1/2) a40 = 11/2 - 39/2 a40 = -28/2 a40 = -14
Arithmetic Progression Applications
Arithmetic progressions have many applications in mathematics and other fields. Some examples include:
- Finance: An arithmetic progression can be used to model the growth of an investment over time.
- Music: An arithmetic progression can be used to create musical scales.
- Science: An arithmetic progression can be used to model the growth of a population over time.
Arithmetic Progression Limitations
Arithmetic progressions have some limitations. For example:
- Assumes constant difference: An arithmetic progression assumes that the difference between consecutive terms is constant, which may not always be the case.
- Does not account for non-linear growth: An arithmetic progression does not account for non-linear growth, which can occur in many real-world situations.
Arithmetic Progression Conclusion
Q: What is an arithmetic progression?
A: An arithmetic progression is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is called the common difference.
Q: What is the formula for the nth term of an arithmetic progression?
A: The formula for the nth term of an arithmetic progression 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: How do I find the first term and the common difference of an arithmetic progression?
A: To find the first term and the common difference of an arithmetic progression, you can use the formula for the nth term. You can write two equations using the formula and solve for the first term and the common difference.
Q: What is the difference between an arithmetic progression and a geometric progression?
A: An arithmetic progression is a sequence of numbers in which the difference between any two consecutive terms is constant. A geometric progression is a sequence of numbers in which the ratio between any two consecutive terms is constant.
Q: Can an arithmetic progression have a negative common difference?
A: Yes, an arithmetic progression can have a negative common difference. In this case, the sequence of numbers will decrease over time.
Q: Can an arithmetic progression have a zero common difference?
A: Yes, an arithmetic progression can have a zero common difference. In this case, the sequence of numbers will be constant over time.
Q: What is the sum of an arithmetic progression?
A: The sum of an arithmetic progression is given by the formula:
S = n/2 (2a + (n - 1)d)
where S is the sum, n is the number of terms, a is the first term, and d is the common difference.
Q: What is the average of an arithmetic progression?
A: The average of an arithmetic progression is given by the formula:
A = (a + l)/2
where A is the average, a is the first term, and l is the last term.
Q: Can an arithmetic progression be used to model real-world situations?
A: Yes, an arithmetic progression can be used to model real-world situations such as the growth of an investment over time, the growth of a population over time, and the creation of musical scales.
Q: What are some limitations of arithmetic progressions?
A: Some limitations of arithmetic progressions include:
- Assumes constant difference: An arithmetic progression assumes that the difference between consecutive terms is constant, which may not always be the case.
- Does not account for non-linear growth: An arithmetic progression does not account for non-linear growth, which can occur in many real-world situations.
Q: How do I determine if a sequence is an arithmetic progression?
A: To determine if a sequence is an arithmetic progression, you can check if the difference between any two consecutive terms is constant. If it is, then the sequence is an arithmetic progression.
Q: Can an arithmetic progression have a fractional common difference?
A: Yes, an arithmetic progression can have a fractional common difference. In this case, the sequence of numbers will increase or decrease by a fraction of the common difference over time.
Q: What is the relationship between arithmetic progressions and quadratic equations?
A: Arithmetic progressions and quadratic equations are related in that the sum of an arithmetic progression can be expressed as a quadratic equation.