In An Arithmetic Progression, The Sum Of The First 5 Terms Is 1550, And The Sum Of The Next 10 Terms Is 325. Find:(i) The First Term And The Common Difference.(ii) The Sum Of The First 13 Terms 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 explore how to find the first term and the common difference of an arithmetic progression given the sum of the first few terms. We will also learn how to find the sum of the first 13 terms of the progression.
The Formula for the Sum of the First n Terms of an Arithmetic Progression
The formula for the sum of the first n terms of an arithmetic progression is given by:
S_n = n/2 * [2a + (n-1)d]
where S_n is the sum of the first n terms, a is the first term, n is the number of terms, and d is the common difference.
Given Information
We are given that the sum of the first 5 terms is 1550 and the sum of the next 10 terms is 325. We can use this information to set up two equations and solve for the first term and the common difference.
Step 1: Setting Up the Equations
Let's denote the first term as a and the common difference as d. We can use the formula for the sum of the first n terms to set up two equations:
- For the first 5 terms: 5/2 * [2a + (5-1)d] = 1550
- For the next 10 terms: 10/2 * [2(a+5d) + (10-1)d] = 325
Simplifying the equations, we get:
- 5a + 10d = 1550
- 10a + 55d = 325
Step 2: Solving the Equations
We can solve the system of equations using substitution or elimination. Let's use the elimination method.
Multiplying the first equation by 2, we get:
10a + 20d = 3100
Now, subtracting the second equation from this, we get:
-45d = 2775
Dividing by -45, we get:
d = -62
Now that we have the common difference, we can substitute it into one of the original equations to solve for the first term.
Substituting d = -62 into the first equation, we get:
5a + 10(-62) = 1550
Simplifying, we get:
5a - 620 = 1550
Adding 620 to both sides, we get:
5a = 2170
Dividing by 5, we get:
a = 434
Conclusion
We have found the first term (a = 434) and the common difference (d = -62) of the arithmetic progression.
The Sum of the First 13 Terms
Now that we have the first term and the common difference, we can find the sum of the first 13 terms using the formula:
S_n = n/2 * [2a + (n-1)d]
Substituting n = 13, a = 434, and d = -62, we get:
S_13 = 13/2 * [2(434) + (13-1)(-62)]
Simplifying, we get:
S_13 = 6.5 * [868 - 806]
S_13 = 6.5 * 62
S_13 = 403
The Final Answer
Therefore, the sum of the first 13 terms of the arithmetic progression is 403.
Discussion
This problem is a classic example of how to use the formula for the sum of the first n terms of an arithmetic progression to solve for the first term and the common difference. The key is to set up the equations correctly and use the elimination method to solve for the common difference.
In this problem, we used the elimination method to solve for the common difference. However, we could have also used the substitution method. The choice of method depends on the specific problem and the values of the variables.
Real-World Applications
Arithmetic progressions have many real-world applications, including:
- Finance: Calculating interest rates and investment returns
- Science: Modeling population growth and decay
- Engineering: Designing systems with periodic behavior
In conclusion, arithmetic progressions are an important concept in mathematics, and understanding how to find the first term and the common difference is crucial for solving problems in various fields.
References
- [1] "Arithmetic Progressions" by Khan Academy
- [2] "Arithmetic Progressions" by Math Open Reference
- [3] "Arithmetic Progressions" by Wolfram MathWorld
Additional Resources
- [1] "Arithmetic Progressions" by MIT OpenCourseWare
- [2] "Arithmetic Progressions" by Stanford University
- [3] "Arithmetic Progressions" by University of California, Berkeley
Frequently Asked Questions
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: 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, you can use the formula for the sum of the first n terms of an arithmetic progression:
S_n = n/2 * [2a + (n-1)d]
where S_n is the sum of the first n terms, a is the first term, n is the number of terms, and d is the common difference.
Q: What is the formula for the sum of the first n terms of an arithmetic progression?
A: The formula for the sum of the first n terms of an arithmetic progression is:
S_n = n/2 * [2a + (n-1)d]
Q: How do I find the sum of the first n terms of an arithmetic progression?
A: To find the sum of the first n terms, you can use the formula:
S_n = n/2 * [2a + (n-1)d]
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: How do I find the nth term of an arithmetic progression?
A: To find the nth term of an arithmetic progression, you can use the formula:
a_n = a + (n-1)d
where a_n is the nth term, a is the first term, n is the term number, and d is the common difference.
Q: What is the sum of the first n terms of an arithmetic progression?
A: The sum of the first n terms of an arithmetic progression is given by the formula:
S_n = n/2 * [2a + (n-1)d]
Q: How do I find the sum of the first n terms of an arithmetic progression when the common difference is not constant?
A: If the common difference is not constant, you cannot use the formula for the sum of the first n terms of an arithmetic progression. In this case, you need to use a different formula or method to find the sum.
Q: What is the formula for the sum of an infinite arithmetic progression?
A: The formula for the sum of an infinite arithmetic progression is:
S = a / (1 - r)
where S is the sum, a is the first term, and r is the common ratio.
Q: How do I find the sum of an infinite arithmetic progression?
A: To find the sum of an infinite arithmetic progression, you can use the formula:
S = a / (1 - r)
Q: What is the difference between an arithmetic progression and a harmonic progression?
A: An arithmetic progression is a sequence of numbers in which the difference between any two consecutive terms is constant. A harmonic progression is a sequence of numbers in which the reciprocals of the terms form an arithmetic progression.
Q: How do I find the nth term of a harmonic progression?
A: To find the nth term of a harmonic progression, you can use the formula:
a_n = 1 / (a + (n-1)d)
where a_n is the nth term, a is the first term, n is the term number, and d is the common difference.
Q: What is the sum of the first n terms of a harmonic progression?
A: The sum of the first n terms of a harmonic progression is given by the formula:
S_n = n/2 * [2a + (n-1)d]
Q: How do I find the sum of the first n terms of a harmonic progression?
A: To find the sum of the first n terms of a harmonic progression, you can use the formula:
S_n = n/2 * [2a + (n-1)d]
Conclusion
Arithmetic progressions are an important concept in mathematics, and understanding how to find the first term and the common difference is crucial for solving problems in various fields. This Q&A article provides answers to frequently asked questions about arithmetic progressions, including how to find the first term and the common difference, how to find the sum of the first n terms, and how to find the nth term.
References
- [1] "Arithmetic Progressions" by Khan Academy
- [2] "Arithmetic Progressions" by Math Open Reference
- [3] "Arithmetic Progressions" by Wolfram MathWorld
Additional Resources
- [1] "Arithmetic Progressions" by MIT OpenCourseWare
- [2] "Arithmetic Progressions" by Stanford University
- [3] "Arithmetic Progressions" by University of California, Berkeley
Note: The references and additional resources provided are for informational purposes only and are not necessarily endorsed by the author.