Which Term Of The Arithmetic Progression 21, 18, 15, Is -81? Also, Find The Term Which Becomes Zero.
Arithmetic progressions are a fundamental concept in mathematics, where each term is obtained by adding or subtracting a fixed constant from the previous term. In this article, we will explore the concept of arithmetic progressions and use it to find the term that is -81 in the given sequence 21, 18, 15, and also determine the term that becomes zero.
What is an Arithmetic Progression?
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. For example, in the sequence 2, 5, 8, 11, 14, the common difference is 3.
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
- d is the common difference
Finding the Term that is -81
To find the term that is -81, we need to use the formula for the nth term of an arithmetic progression. We are given the sequence 21, 18, 15, and we need to find the term that is -81.
First, let's find the common difference (d) between the given terms:
d = 18 - 21 = -3
Now, we can use the formula for the nth term to find the term that is -81:
-81 = 21 + (n - 1)(-3)
Simplifying the equation, we get:
-81 = 21 - 3n + 3
Combine like terms:
-81 = 24 - 3n
Subtract 24 from both sides:
-105 = -3n
Divide both sides by -3:
35 = n
Therefore, the term that is -81 is the 35th term.
Finding the Term that Becomes Zero
To find the term that becomes zero, we need to find the value of n for which the nth term is zero. We can use the formula for the nth term to find this value:
0 = 21 + (n - 1)(-3)
Simplifying the equation, we get:
0 = 21 - 3n + 3
Combine like terms:
0 = 24 - 3n
Subtract 24 from both sides:
-24 = -3n
Divide both sides by -3:
8 = n
Therefore, the term that becomes zero is the 8th term.
Conclusion
In this article, we used the concept of arithmetic progressions to find the term that is -81 in the given sequence 21, 18, 15, and also determined the term that becomes zero. We used the formula for the nth term of an arithmetic progression to solve these problems. The common difference between the given terms was found to be -3, and the term that is -81 was found to be the 35th term. The term that becomes zero was found to be the 8th term.
Arithmetic Progression Examples
Here are some examples of arithmetic progressions:
- 2, 5, 8, 11, 14 (common difference = 3)
- 10, 7, 4, 1, -2 (common difference = -3)
- 15, 12, 9, 6, 3 (common difference = -3)
Arithmetic Progression Formula
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
- d is the common difference
Arithmetic Progression Applications
Arithmetic progressions have many applications in real-life situations, such as:
- Finance: calculating interest rates and investment returns
- Science: modeling population growth and decay
- Engineering: designing electrical circuits and mechanical systems
Arithmetic Progression Exercises
Here are some exercises to practice arithmetic progressions:
- Find the 10th term of the arithmetic progression 2, 5, 8, 11, 14.
- Find the term that becomes zero in the arithmetic progression 10, 7, 4, 1, -2.
- Find the common difference of the arithmetic progression 15, 12, 9, 6, 3.
Arithmetic Progression Solutions
Here are the solutions to the exercises:
- The 10th term of the arithmetic progression 2, 5, 8, 11, 14 is 29.
- The term that becomes zero in the arithmetic progression 10, 7, 4, 1, -2 is the 5th term.
- The common difference of the arithmetic progression 15, 12, 9, 6, 3 is -3.
Arithmetic Progression Q&A =============================
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 given by:
an = a + (n - 1)d
where:
- an is the nth term
- a is the first term
- n is the term number
- d is the common difference
Q: How do I find the common difference of an arithmetic progression?
A: To find the common difference of an arithmetic progression, you can subtract any two consecutive terms. For example, if the sequence is 2, 5, 8, 11, 14, you can subtract the first term from the second term to get the common difference: 5 - 2 = 3.
Q: How do I find the term number of an arithmetic progression?
A: To find the term number of an arithmetic progression, you can use the formula for the nth term:
an = a + (n - 1)d
Rearrange the formula to solve for n:
n = (an - a) / d + 1
Q: What is the relationship between the first term, common difference, and term number in an arithmetic progression?
A: The relationship between the first term, common difference, and term number in an arithmetic progression is given by the formula:
an = a + (n - 1)d
This formula shows that the nth term is equal to the first term plus the product of the common difference and the term number minus one.
Q: Can an arithmetic progression have a zero term?
A: Yes, an arithmetic progression can have a zero term. To find the term number of the zero term, you can use the formula:
an = a + (n - 1)d
Set the nth term equal to zero and solve for n:
0 = a + (n - 1)d
Solve for n:
n = (a / d) + 1
Q: Can an arithmetic progression have a negative term?
A: Yes, an arithmetic progression can have a negative term. To find the term number of the negative term, you can use the formula:
an = a + (n - 1)d
Set the nth term equal to the negative value and solve for n:
an = a + (n - 1)d
Solve for n:
n = ((a - an) / d) + 1
Q: What are some real-life applications of arithmetic progressions?
A: Arithmetic progressions have many real-life applications, such as:
- Finance: calculating interest rates and investment returns
- Science: modeling population growth and decay
- Engineering: designing electrical circuits and mechanical systems
Q: How do I find the sum of an arithmetic progression?
A: To find the sum of an arithmetic progression, you can use the formula:
S = (n / 2)(a + l)
where:
- S is the sum
- n is the number of terms
- a is the first term
- l is the last term
Q: How do I find the average of an arithmetic progression?
A: To find the average of an arithmetic progression, you can use the formula:
A = (a + l) / 2
where:
- A is the average
- a is the first term
- l is the last term
Q: Can an arithmetic progression have a variable common difference?
A: Yes, an arithmetic progression can have a variable common difference. In this case, the sequence is called a quadratic progression.
Q: Can an arithmetic progression have a variable first term?
A: Yes, an arithmetic progression can have a variable first term. In this case, the sequence is called a linear progression.
Q: What is the relationship between arithmetic progressions and geometric progressions?
A: Arithmetic progressions and geometric progressions are two different types of sequences. While arithmetic progressions have a constant difference between terms, geometric progressions have a constant ratio between terms.