Relationship Between First Term and Common Difference

Oct 13, 2025 by windailys.com 20 views

Answer: C8. Relationship Between First Term and Common Difference in an Arithmetic Progression

Understanding the Problem

In an arithmetic progression (AP), each term is obtained by adding a fixed constant to the previous term. 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.

Given Information

We are given that the ninth term of the AP is five times the fifth term. Mathematically, this can be expressed as:

$a + 8d = 5(a + 4d)$

Simplifying the Equation

To find the relationship between $a$ and $d$ , we need to simplify the equation obtained above.

$a + 8d = 5a + 20d$

Subtracting $a$ from both sides:

$8d = 4a + 20d$

Subtracting $20d$ from both sides:

$-12d = 4a$

Dividing both sides by $-12$ :

$d = -\frac{4}{12}a$

$d = -\frac{1}{3}a$

Relationship Between First Term and Common Difference

From the equation obtained above, we can see that the common difference $d$ is equal to $-\frac{1}{3}$ times the first term $a$ . This means that if the first term is positive, the common difference will be negative, and vice versa.

Conclusion

In this problem, we were given that the ninth term of an AP is five times the fifth term. By simplifying the equation obtained from this information, we found that the common difference $d$ is equal to $-\frac{1}{3}$ times the first term $a$ . This relationship between the first term and the common difference is a fundamental property of arithmetic progressions.

Example Use Case

Suppose we have an AP with a first term of $10$ and a common difference of $-3$ . Using the relationship obtained above, we can verify that the ninth term is indeed five times the fifth term.

$a = 10$ $d = -3$

Fifth term:

$a + 4d = 10 + 4(-3) = 10 - 12 = -2$

Ninth term:

$a + 8d = 10 + 8(-3) = 10 - 24 = -14$

Since $-14 = 5(-2)$ , we can see that the ninth term is indeed five times the fifth term.

Key Takeaways

The relationship between the first term and the common difference in an arithmetic progression is given by $d = -\frac{1}{3}a$ .
If the first term is positive, the common difference will be negative, and vice versa.
This relationship can be used to verify the properties of arithmetic progressions.

Further Reading

For more information on arithmetic progressions, including their properties and applications, please refer to the following resources:

Arithmetic Progression
Arithmetic Progression Formula
Arithmetic Progression Examples
Q&A: Relationship Between First Term and Common Difference in an Arithmetic Progression

Q1: What is the relationship between the first term and the common difference in an arithmetic progression?

A1: The relationship between the first term and the common difference in an arithmetic progression is given by $d = -\frac{1}{3}a$ . This means that the common difference $d$ is equal to $-\frac{1}{3}$ times the first term $a$ .

Q2: How can I verify the properties of an arithmetic progression using the relationship between the first term and the common difference?

A2: You can verify the properties of an arithmetic progression by using the relationship between the first term and the common difference. For example, if you know the first term and the common difference, you can use the formula to find the nth term of the progression.

Q3: What happens if the first term is positive? Does the common difference change sign?

A3: Yes, if the first term is positive, the common difference will be negative, and vice versa. This is because the relationship between the first term and the common difference is given by $d = -\frac{1}{3}a$ . If $a$ is positive, then $d$ will be negative, and if $a$ is negative, then $d$ will be positive.

Q4: Can I use the relationship between the first term and the common difference to find the nth term of an arithmetic progression?

A4: Yes, you can use the relationship between the first term and the common difference to find the nth term of an arithmetic progression. The formula for the nth term of an arithmetic progression is given by: $a_n = a + (n-1)d$ . You can substitute the value of $d$ from the relationship between the first term and the common difference into this formula to find the nth term.

Q5: What are some real-world applications of arithmetic progressions?

A5: Arithmetic progressions have many real-world applications, including:

Finance: Arithmetic progressions are used to calculate interest rates and investment returns.
Music: Arithmetic progressions are used to create musical scales and melodies.
Science: Arithmetic progressions are used to model population growth and decay.
Engineering: Arithmetic progressions are used to design and optimize systems.

Q6: Can I use the relationship between the first term and the common difference to find the sum of an arithmetic progression?

A6: Yes, you can use the relationship between the first term and the common difference to find the sum of an arithmetic progression. The formula for the sum of an arithmetic progression is given by: $S_n = \frac{n}{2}(a + a_n)$ . You can substitute the value of $a_n$ from the relationship between the first term and the common difference into this formula to find the sum.

Q7: What are some common mistakes to avoid when working with arithmetic progressions?

A7: Some common mistakes to avoid when working with arithmetic progressions include:

Assuming that the common difference is always positive.
Failing to check the sign of the common difference.
Using the wrong formula for the nth term or sum of an arithmetic progression.

Q8: Can I use the relationship between the first term and the common difference to find the median of an arithmetic progression?

A8: Yes, you can use the relationship between the first term and the common difference to find the median of an arithmetic progression. The median of an arithmetic progression is given by: $M = a + \frac{n-1}{2}d$ . You can substitute the value of $d$ from the relationship between the first term and the common difference into this formula to find the median.

Q9: What are some advanced topics in arithmetic progressions?

A9: Some advanced topics in arithmetic progressions include:

Geometric progressions: These are progressions where each term is obtained by multiplying the previous term by a fixed constant.
Harmonic progressions: These are progressions where each term is obtained by taking the reciprocal of the previous term.
Recursive progressions: These are progressions where each term is obtained by applying a recursive formula.

Q10: Can I use the relationship between the first term and the common difference to find the mode of an arithmetic progression?

A10: No, you cannot use the relationship between the first term and the common difference to find the mode of an arithmetic progression. The mode of an arithmetic progression is the term that appears most frequently, and this cannot be determined from the relationship between the first term and the common difference.