The 11th and 13th Terms of an AP

Oct 13, 2025 by leolavanderia.com.br 33 views

The Common Difference of an Arithmetic Progression (AP)

Understanding 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 known as the common difference. 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 11th and 13th Terms of an AP

We are given that the 11th term of an AP is 39 and the 13th term is 45. We can use this information to find the common difference of the AP.

Using the Formula for the nth Term of an AP

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

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.

Finding the Common Difference

We can use the given information to set up two equations using the formula for the nth term of an AP:

For the 11th term: 39 = a + (11 - 1)d 39 = a + 10d

For the 13th term: 45 = a + (13 - 1)d 45 = a + 12d

Solving the Equations

We can solve these two equations simultaneously to find the values of 'a' and 'd'.

Subtracting the first equation from the second equation, we get:

45 - 39 = (a + 12d) - (a + 10d) 6 = 2d

Finding the Common Difference

Dividing both sides of the equation by 2, we get:

d = 6/2 d = 3

Therefore, the common difference of the AP is 3.

Conclusion

In this article, we have discussed the concept of arithmetic progression (AP) and how to find the common difference of an AP using the formula for the nth term. We have used the given information about the 11th and 13th terms of an AP to set up two equations and solve them simultaneously to find the common difference. The common difference of the AP is 3.

References

[1] Khan Academy. (n.d.). Arithmetic Progression. Retrieved from https://www.khanacademy.org/math/algebra2/x2f4f4d/x2f4f4d/x2f4f4d
[2] Math Open Reference. (n.d.). Arithmetic Progression. Retrieved from https://www.mathopenref.com/arithmeticprogression.html

Discussion

What is the common difference of an arithmetic progression (AP) if the 11th term is 39 and the 13th term is 45?

A. 42 B. 21 C. 6 D. 3

Answer

The correct answer is D. 3.
Q&A: Arithmetic Progression (AP)

Understanding Arithmetic Progression (AP)

In our previous article, we discussed the concept of arithmetic progression (AP) and how to find the common difference of an AP using the formula for the nth term. In this article, we will answer some frequently asked questions about AP.

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 known as the common difference.

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

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

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 of an AP?

A: To find the common difference of an AP, you can use the formula for the nth term and set up two equations using the given information. For example, if the 11th term is 39 and the 13th term is 45, you can set up the following equations:

39 = a + 10d 45 = a + 12d

Subtracting the first equation from the second equation, you get:

6 = 2d

Dividing both sides of the equation by 2, you get:

d = 3

Therefore, the common difference of the AP is 3.

Q: What is the relationship between the first term and the common difference in an AP?

A: The first term and the common difference are related by the formula for the nth term. If you know the first term and the common difference, you can find any term in the AP using the formula.

Q: Can I find the common difference of an AP if I know only one term?

A: No, you cannot find the common difference of an AP if you know only one term. You need at least two terms to find the common difference.

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

A: The sum of the first n terms of an AP is given by the formula:

Sn = n/2 (a + l)

where 'Sn' is the sum of the first n terms, 'n' is the number of terms, 'a' is the first term, and 'l' is the last term.

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

A: To find the sum of the first n terms of an AP, you need to know the first term, the last term, and the number of terms. You can use the formula:

Sn = n/2 (a + l)

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

A: The formula for the sum of an infinite AP is given by:

S = a / (1 - r)

where 'S' is the sum of the infinite AP, 'a' is the first term, and 'r' is the common ratio.

Q: How do I find the sum of an infinite AP?

A: To find the sum of an infinite AP, you need to know the first term and the common ratio. You can use the formula:

S = a / (1 - r)

Conclusion

In this article, we have answered some frequently asked questions about arithmetic progression (AP). We have discussed the concept of AP, the formula for the nth term, and how to find the common difference, sum of the first n terms, and sum of an infinite AP.

References

[1] Khan Academy. (n.d.). Arithmetic Progression. Retrieved from https://www.khanacademy.org/math/algebra2/x2f4f4d/x2f4f4d/x2f4f4d
[2] Math Open Reference. (n.d.). Arithmetic Progression. Retrieved from https://www.mathopenref.com/arithmeticprogression.html

Discussion

Do you have any questions about arithmetic progression (AP)? Ask us in the comments below!