Which Term Of The Arithmetic Progression (AP) \[$21, 18, 15, \ldots\$\] Is \[$-81\$\]?

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Introduction

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 explore how to find the term in an arithmetic progression given a specific value. We will use the given AP ${21, 18, 15, \ldots\$} and find the term {-81$}$.

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. For example, in the AP ${21, 18, 15, \ldots\$}, the common difference is −3${-3\$}.

Formula for the nth Term of an AP

The formula for the nth term of an AP is given by:

an=a1+(n−1)da_n = a_1 + (n-1)d

where ana_n is the nth term, a1a_1 is the first term, nn is the term number, and dd is the common difference.

Finding the Term in the AP

We are given the AP ${21, 18, 15, \ldots\$} and we need to find the term {-81$}$. We can use the formula for the nth term of an AP to solve for the term number nn.

Let's substitute the values into the formula:

−81=21+(n−1)(−3)-81 = 21 + (n-1)(-3)

Simplifying the equation, we get:

−81=21−3n+3-81 = 21 - 3n + 3

Combine like terms:

−81=24−3n-81 = 24 - 3n

Subtract 24 from both sides:

−105=−3n-105 = -3n

Divide both sides by -3:

n=35n = 35

Conclusion

In this article, we have explored how to find the term in an arithmetic progression given a specific value. We used the given AP ${21, 18, 15, \ldots\$} and found the term {-81$}$. We used the formula for the nth term of an AP to solve for the term number nn. The term number nn is 35.

Example Use Cases

  1. Find the term in the AP ${2, 5, 8, \ldots\$} that is equal to ${17\$}.
  2. Find the term in the AP ${10, 12, 14, \ldots\$} that is equal to ${32\$}.

Step-by-Step Solution

  1. Identify the first term, common difference, and the term you want to find.
  2. Use the formula for the nth term of an AP to set up an equation.
  3. Solve for the term number nn.
  4. Use the term number nn to find the term in the AP.

Common Mistakes

  1. Not identifying the first term and common difference correctly.
  2. Not using the correct formula for the nth term of an AP.
  3. Not solving for the term number nn correctly.

Tips and Tricks

  1. Make sure to identify the first term and common difference correctly.
  2. Use the correct formula for the nth term of an AP.
  3. Solve for the term number nn carefully.

Conclusion

Q: What is an arithmetic progression?

A: An arithmetic progression (AP) 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 common difference in an arithmetic progression?

A: To find the common difference, subtract any term from its previous term. For example, in the AP ${21, 18, 15, \ldots\$}, the common difference is −3${-3\$} because 18−21=−3${18 - 21 = -3\$}.

Q: What is the formula for the nth term of an arithmetic progression?

A: The formula for the nth term of an AP is given by:

an=a1+(n−1)da_n = a_1 + (n-1)d

where ana_n is the nth term, a1a_1 is the first term, nn is the term number, and dd is the common difference.

Q: How do I find the term number in an arithmetic progression?

A: To find the term number, use the formula for the nth term of an AP and solve for nn. For example, if we want to find the term number in the AP ${21, 18, 15, \ldots\$} that is equal to {-81$}$, we can use the formula:

−81=21+(n−1)(−3)-81 = 21 + (n-1)(-3)

Solving for nn, we get:

n=35n = 35

Q: What is the significance of the term number in an arithmetic progression?

A: The term number is the position of the term in the arithmetic progression. It is used to identify the term and to find the value of the term.

Q: Can I use the formula for the nth term of an arithmetic progression to find the first term?

A: Yes, you can use the formula for the nth term of an AP to find the first term. If you know the value of the nth term, the common difference, and the term number, you can solve for the first term.

Q: What are some common mistakes to avoid when working with arithmetic progressions?

A: Some common mistakes to avoid when working with arithmetic progressions include:

  • Not identifying the first term and common difference correctly
  • Not using the correct formula for the nth term of an AP
  • Not solving for the term number correctly

Q: How do I apply the concept of arithmetic progression in real-life situations?

A: Arithmetic progressions are used in many real-life situations, such as:

  • Finance: calculating interest rates and investment returns
  • Science: modeling population growth and decay
  • Engineering: designing and optimizing systems

Q: Can I use arithmetic progressions to solve problems in other areas of mathematics?

A: Yes, arithmetic progressions can be used to solve problems in other areas of mathematics, such as:

  • Algebra: solving linear equations and inequalities
  • Geometry: finding the perimeter and area of shapes
  • Trigonometry: solving triangles and finding angles

Q: What are some advanced topics related to arithmetic progressions?

A: Some advanced topics related to arithmetic progressions include:

  • Geometric progressions: a sequence of numbers in which the ratio between any two consecutive terms is constant
  • Harmonic progressions: a sequence of numbers in which the reciprocals of the terms form an arithmetic progression
  • Infinite arithmetic progressions: an arithmetic progression that has an infinite number of terms