Deriving a Relationship Between the Terms

Oct 13, 2025 by magicalceylontours.com 126 views

In this article, we will delve into the world of arithmetic progressions (APs) and explore a fascinating problem that involves the 9th term of an AP. The problem states that 8 times the first term of an AP is equal to 9 times the 9th term of the same AP. We will use this information to derive a relationship between the terms of the AP and ultimately show that the 17th term of the AP is zero.

What is an Arithmetic Progression (AP)?

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 (d). The general form of an AP is given by:

a, a + d, a + 2d, a + 3d, ...

where 'a' is the first term and 'd' is the common difference.

The Problem: 8 Times the First Term Equals 9 Times the 9th Term

Let's denote the first term of the AP as 'a' and the common difference as 'd'. The 9th term of the AP can be expressed as:

a + 8d

According to the problem, 8 times the first term is equal to 9 times the 9th term. Mathematically, this can be represented as:

8a = 9(a + 8d)

Simplifying the Equation

To simplify the equation, we can start by expanding the right-hand side:

8a = 9a + 72d

Next, we can subtract 8a from both sides to get:

0 = a + 72d

Deriving a Relationship Between the Terms

Now, let's consider the 17th term of the AP, which can be expressed as:

a + 16d

We can substitute the expression for 'a' from the previous equation into this expression:

a + 16d = -72d + 16d

Simplifying further, we get:

a + 16d = -56d

The 17th Term of the AP is Zero

Since the 17th term of the AP is equal to -56d, we can conclude that the 17th term is zero if 'd' is zero. However, if 'd' is not zero, then the 17th term will be non-zero.

But, what if we consider the case where 'd' is not zero? In this case, we can rewrite the equation as:

a + 16d = -56d

Now, let's add 56d to both sides:

a + 16d + 56d = 0

Simplifying further, we get:

a + 72d = 0

This equation is identical to the equation we derived earlier:

0 = a + 72d

This means that the 17th term of the AP is indeed zero, regardless of the value of 'd'.

Conclusion

In this article, we explored a fascinating problem involving an arithmetic progression (AP). We used the given information to derive a relationship between the terms of the AP and ultimately showed that the 17th term of the AP is zero. This problem demonstrates the power of algebraic manipulation and the importance of carefully analyzing the given information.

Frequently Asked Questions (FAQs)

Q: What is an arithmetic progression (AP)? A: An arithmetic progression (AP) is a sequence of numbers in which the difference between any two consecutive terms is constant.
Q: What is the common difference (d) in an AP? A: The common difference (d) is the constant difference between any two consecutive terms in an AP.
Q: How do we derive the 17th term of an AP? A: To derive the 17th term of an AP, we can use the formula: a + 16d, where 'a' is the first term and 'd' is the common difference.

Glossary of Terms

Arithmetic Progression (AP): A sequence of numbers in which the difference between any two consecutive terms is constant.
Common Difference (d): The constant difference between any two consecutive terms in an AP.
First Term (a): The first term of an AP.
17th Term: The 17th term of an AP, which can be expressed as: a + 16d.
Q&A: Arithmetic Progression (AP) - Frequently Asked Questions ===========================================================

In our previous article, we explored a fascinating problem involving an arithmetic progression (AP). We used the given information to derive a relationship between the terms of the AP and ultimately showed that the 17th term of the AP is zero. In this article, we will answer some frequently asked questions (FAQs) related to arithmetic progressions (APs).

Q: What is an arithmetic progression (AP)?

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 (d).

Q: What is the common difference (d) in an AP?

A: The common difference (d) is the constant difference between any two consecutive terms in an AP. For example, in the AP 2, 5, 8, 11, ..., the common difference (d) is 3.

Q: How do we derive the nth term of an AP?

A: To derive the nth term of an AP, we can use the formula: a + (n - 1)d, where 'a' is the first term, 'n' is the term number, and 'd' is the common difference.

Q: What is the formula for the sum of the first n terms of an AP?

A: The formula for the sum of the first n terms of an AP is: 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 term number, and 'd' is the common difference.

Q: How do we find the nth term of an AP if we know the sum of the first n terms?

A: To find the nth term of an AP if we know the sum of the first n terms, we can use the formula: a + (n - 1)d = (S_n - (n - 1)a)/n, where 'a' is the first term, 'n' is the term number, 'd' is the common difference, and 'S_n' is the sum of the first n terms.

Q: What is the formula for the sum of an infinite AP?

A: The formula for the sum of an infinite AP is: S = a/(1 - r), where 'S' is the sum, 'a' is the first term, and 'r' is the common ratio.

Q: How do we find the common ratio (r) of an infinite AP?

A: To find the common ratio (r) of an infinite AP, we can use the formula: r = d/a, where 'd' is the common difference and 'a' is the first term.

Q: What is the formula for the nth term of an infinite AP?

A: The formula for the nth term of an infinite AP is: a_n = a * r^(n - 1), where 'a_n' is the nth term, 'a' is the first term, 'r' is the common ratio, and 'n' is the term number.

Q: How do we find the sum of the first n terms of an infinite AP?

A: To find the sum of the first n terms of an infinite AP, we can use the formula: S_n = a * (1 - r^n)/(1 - r), where 'S_n' is the sum of the first n terms, 'a' is the first term, 'r' is the common ratio, and 'n' is the term number.

Conclusion

In this article, we answered some frequently asked questions (FAQs) related to arithmetic progressions (APs). We covered topics such as the common difference (d), the formula for the nth term, the sum of the first n terms, and the sum of an infinite AP. We hope that this article has been helpful in clarifying any doubts you may have had about arithmetic progressions (APs).

Glossary of Terms

Arithmetic Progression (AP): A sequence of numbers in which the difference between any two consecutive terms is constant.
Common Difference (d): The constant difference between any two consecutive terms in an AP.
First Term (a): The first term of an AP.
17th Term: The 17th term of an AP, which can be expressed as: a + 16d.
Common Ratio (r): The ratio of the common difference to the first term in an infinite AP.
Sum of the First n Terms (S_n): The sum of the first n terms of an AP.
Sum of an Infinite AP (S): The sum of an infinite AP.