Find The Term Of An Arithmetic Progression (AP) \[$3 \frac{1}{2}, 7, 10^{3 / 2}\$\] Which Is 77.
Finding the Term of an Arithmetic Progression (AP)
An arithmetic progression (AP) is a sequence of numbers in which the difference between any two consecutive terms is constant. In this article, we will discuss how to find the term of an arithmetic progression given a specific term and the common difference. We will use the given AP ${$3 \frac{1}{2}, 7, 10^{3 / 2}$] and find the term that is equal to 77.
Understanding 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. The general form of an arithmetic progression is:
a, a + d, a + 2d, a + 3d, ...
where a is the first term and d is the common difference.
Finding the Term of an Arithmetic Progression
To find the term of an arithmetic progression, we can use the formula:
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 AP and Term Number
The given AP is [$3 \frac{1}{2}, 7, 10^{3 / 2}$] and the term number is 77. We need to find the term that is equal to 77.
Step 1: Convert the Mixed Number to an Improper Fraction
The first term of the AP is [$3 \frac{1}{2}$. We need to convert this mixed number to an improper fraction.
Step 2: Find the Common Difference
The second term of the AP is 7. We can find the common difference by subtracting the first term from the second term.
d = 7 - \frac{7}{2} d = \frac{14}{2} - \frac{7}{2} d = \frac{7}{2}
Step 3: Find the Term Number
We are given that the term number is 77. However, we need to find the term number that corresponds to the term 77.
Let's assume that the term number is n. We can use the formula:
an = a + (n - 1)d
Substituting the values, we get:
77 = \frac{7}{2} + (n - 1)\frac{7}{2}
Simplifying the equation, we get:
77 = \frac{7}{2} + \frac{7n}{2} - \frac{7}{2}
Combine like terms:
77 = \frac{7n}{2}
Multiply both sides by 2:
154 = 7n
Divide both sides by 7:
n = 22
Step 4: Find the Term
Now that we have the term number, we can find the term using the formula:
an = a + (n - 1)d
Substituting the values, we get:
an = \frac{7}{2} + (22 - 1)\frac{7}{2}
Simplifying the equation, we get:
an = \frac{7}{2} + 21\frac{7}{2}
Combine like terms:
an = \frac{7}{2} + \frac{147}{2}
an = \frac{154}{2}
an = 77
In this article, we discussed how to find the term of an arithmetic progression given a specific term and the common difference. We used the given AP [$3 \frac{1}{2}, 7, 10^{3 / 2}$] and found the term that is equal to 77. We converted the mixed number to an improper fraction, found the common difference, and then found the term number. Finally, we found the term using the formula.
The formula for 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.
The formula for the common difference is:
d = an - a
where an is the nth term and a is the first term.
The formula for the term number is:
n = (an - a) / d
where an is the nth term, a is the first term, and d is the common difference.
- Find the term of the arithmetic progression [2, 5, 8, 11, 14}$ that is equal to 21.
- Find the term number of the arithmetic progression ${3, 6, 9, 12, 15}$ that corresponds to the term 24.
- Find the term of the arithmetic progression ${1, 4, 7, 10, 13}$ that is equal to 36.
- The common difference is 3. We can use the formula:
an = a + (n - 1)d
Substituting the values, we get:
21 = 2 + (n - 1)3
Simplifying the equation, we get:
21 = 2 + 3n - 3
Combine like terms:
21 = 3n - 1
Add 1 to both sides:
22 = 3n
Divide both sides by 3:
n = 22/3
n = 7.33
Since the term number must be an integer, we round up to the nearest integer:
n = 8
Now that we have the term number, we can find the term using the formula:
an = a + (n - 1)d
Substituting the values, we get:
an = 2 + (8 - 1)3
Simplifying the equation, we get:
an = 2 + 7*3
Combine like terms:
an = 2 + 21
an = 23
- The common difference is 3. We can use the formula:
an = a + (n - 1)d
Substituting the values, we get:
24 = 3 + (n - 1)3
Simplifying the equation, we get:
24 = 3 + 3n - 3
Combine like terms:
24 = 3n
Divide both sides by 3:
n = 24/3
n = 8
- The common difference is 3. We can use the formula:
an = a + (n - 1)d
Substituting the values, we get:
36 = 1 + (n - 1)3
Simplifying the equation, we get:
36 = 1 + 3n - 3
Combine like terms:
36 = 3n - 2
Add 2 to both sides:
38 = 3n
Divide both sides by 3:
n = 38/3
n = 12.67
Since the term number must be an integer, we round up to the nearest integer:
n = 13
Now that we have the term number, we can find the term using the formula:
an = a + (n - 1)d
Substituting the values, we get:
an = 1 + (13 - 1)3
Simplifying the equation, we get:
an = 1 + 12*3
Combine like terms:
an = 1 + 36
an = 37
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 an arithmetic progression?
A: The formula for 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 common difference?
A: To find the common difference, you can subtract the first term from the second term.
d = an - a
Q: How do I find the term number?
A: To find the term number, you can use the formula:
n = (an - a) / d
where an is the nth term, a is the first term, and d is the common difference.
Q: What is the term number formula used for?
A: The term number formula is used to find the term number that corresponds to a specific term in an arithmetic progression.
Q: Can I use the term number formula to find the term?
A: Yes, you can use the term number formula to find the term. Once you have the term number, you can use the formula:
an = a + (n - 1)d
to find the term.
Q: What is the relationship between the term number and the term?
A: The term number and the term are related by the formula:
an = a + (n - 1)d
This formula shows that the term is equal to the first term plus the product of the common difference and the term number minus one.
Q: Can I use the arithmetic progression formula to find the common difference?
A: Yes, you can use the arithmetic progression formula to find the common difference. Once you have the term number and the term, you can use the formula:
d = (an - a) / (n - 1)
to find the common difference.
Q: What is the relationship between the common difference and the term number?
A: The common difference and the term number are related by the formula:
d = (an - a) / (n - 1)
This formula shows that the common difference is equal to the difference between the term and the first term divided by the term number minus one.
Q: Can I use the arithmetic progression formula to find the first term?
A: Yes, you can use the arithmetic progression formula to find the first term. Once you have the term number and the term, you can use the formula:
a = an - (n - 1)d
to find the first term.
Q: What is the relationship between the first term and the term number?
A: The first term and the term number are related by the formula:
a = an - (n - 1)d
This formula shows that the first term is equal to the term minus the product of the common difference and the term number minus one.
Q: Can I use the arithmetic progression formula to find the term number?
A: Yes, you can use the arithmetic progression formula to find the term number. Once you have the term and the common difference, you can use the formula:
n = (an - a) / d + 1
to find the term number.
Q: What is the relationship between the term number and the common difference?
A: The term number and the common difference are related by the formula:
n = (an - a) / d + 1
This formula shows that the term number is equal to the difference between the term and the first term divided by the common difference plus one.
Q: Can I use the arithmetic progression formula to find the term?
A: Yes, you can use the arithmetic progression formula to find the term. Once you have the term number and the common difference, you can use the formula:
an = a + (n - 1)d
to find the term.
Q: What is the relationship between the term and the common difference?
A: The term and the common difference are related by the formula:
an = a + (n - 1)d
This formula shows that the term is equal to the first term plus the product of the common difference and the term number minus one.
Q: Can I use the arithmetic progression formula to find the first term?
A: Yes, you can use the arithmetic progression formula to find the first term. Once you have the term number and the term, you can use the formula:
a = an - (n - 1)d
to find the first term.
Q: What is the relationship between the first term and the term number?
A: The first term and the term number are related by the formula:
a = an - (n - 1)d
This formula shows that the first term is equal to the term minus the product of the common difference and the term number minus one.
In this article, we have discussed the arithmetic progression formula and its applications. We have also provided answers to frequently asked questions about arithmetic progressions. We hope that this article has been helpful in understanding the concept of arithmetic progressions and how to use the formula to find the term, common difference, and first term.