If D 6th Term Of An AP Is 11,dd First Term Is 1.Find D

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If the 6th Term of an AP is 11 and the First Term is 1, Find the Common Difference 'd'

Understanding the Basics of 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, denoted by 'd'. The general form of an AP is given by: a, a + d, a + 2d, a + 3d, ...

Given Information

We are given that the 6th term of the AP is 11 and the first term is 1. We need to find the common difference 'd'.

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 'a' is the first term, 'n' is the term number, and 'd' is the common difference.

Applying the Formula to the Given Information

We are given that the 6th term is 11 and the first term is 1. We can use the formula to write: 11 = 1 + (6 - 1)d.

Simplifying the Equation

Simplifying the equation, we get: 11 = 1 + 5d.

Solving for 'd'

Subtracting 1 from both sides, we get: 10 = 5d.

Dividing both sides by 5, we get: d = 2.

Conclusion

Therefore, the common difference 'd' is 2.

Example and Practice Problems

Here are a few example and practice problems to help you understand the concept better:

  • If the 8th term of an AP is 27 and the first term is 3, find the common difference 'd'.
  • If the 4th term of an AP is 13 and the first term is 2, find the common difference 'd'.
  • If the 10th term of an AP is 41 and the first term is 5, find the common difference 'd'.

Solution to Example Problems

  • If the 8th term of an AP is 27 and the first term is 3, we can use the formula to write: 27 = 3 + (8 - 1)d.

  • Simplifying the equation, we get: 27 = 3 + 7d.

  • Subtracting 3 from both sides, we get: 24 = 7d.

  • Dividing both sides by 7, we get: d = 24/7.

  • Therefore, the common difference 'd' is 24/7.

  • If the 4th term of an AP is 13 and the first term is 2, we can use the formula to write: 13 = 2 + (4 - 1)d.

  • Simplifying the equation, we get: 13 = 2 + 3d.

  • Subtracting 2 from both sides, we get: 11 = 3d.

  • Dividing both sides by 3, we get: d = 11/3.

  • Therefore, the common difference 'd' is 11/3.

  • If the 10th term of an AP is 41 and the first term is 5, we can use the formula to write: 41 = 5 + (10 - 1)d.

  • Simplifying the equation, we get: 41 = 5 + 9d.

  • Subtracting 5 from both sides, we get: 36 = 9d.

  • Dividing both sides by 9, we get: d = 36/9.

  • Therefore, the common difference 'd' is 4.

Tips and Tricks

  • To find the common difference 'd', you can use the formula: d = (an - a) / (n - 1), where 'an' is the nth term, 'a' is the first term, and 'n' is the term number.
  • You can also use the formula: d = (an - a1) / (n - 1), where 'an' is the nth term, 'a1' is the first term, and 'n' is the term number.
  • Make sure to simplify the equation and solve for 'd' carefully.

Conclusion

In this article, we learned how to find the common difference 'd' of an arithmetic progression (AP) given the first term and the nth term. We used the formula for the nth term of an AP and simplified the equation to solve for 'd'. We also provided example and practice problems to help you understand the concept better.
Q&A: Arithmetic Progression (AP) - Common Difference 'd'

Frequently Asked Questions

Here are some frequently asked questions about arithmetic progression (AP) and common difference 'd':

Q1: What is the common difference 'd' in an arithmetic progression (AP)?

A1: The common difference 'd' is the constant difference between any two consecutive terms in an arithmetic progression (AP).

Q2: How do I find the common difference 'd' given the first term and the nth term?

A2: You can use the formula: d = (an - a) / (n - 1), where 'an' is the nth term, 'a' is the first term, and 'n' is the term number.

Q3: What if I don't know the term number 'n'?

A3: You can use the formula: d = (an - a1) / (n - 1), where 'an' is the nth term, 'a1' is the first term, and 'n' is the term number.

Q4: How do I simplify the equation to solve for 'd'?

A4: You can simplify the equation by subtracting the first term from both sides and then dividing both sides by the term number minus one.

Q5: What if the equation is not in the form (an - a) / (n - 1)?

A5: You can rearrange the equation to get it in the form (an - a) / (n - 1) and then solve for 'd'.

Q6: Can I use the formula d = (an - a) / (n - 1) for any value of 'n'?

A6: Yes, you can use the formula for any value of 'n', but make sure to simplify the equation and solve for 'd' carefully.

Q7: What if I get a negative value for 'd'?

A7: If you get a negative value for 'd', it means that the terms are decreasing, not increasing.

Q8: Can I use the formula d = (an - a1) / (n - 1) for any value of 'n'?

A8: Yes, you can use the formula for any value of 'n', but make sure to simplify the equation and solve for 'd' carefully.

Q9: How do I check if the given information is correct?

A9: You can check if the given information is correct by plugging the values into the formula and simplifying the equation.

Q10: What if I get a different value for 'd' using different formulas?

A10: If you get a different value for 'd' using different formulas, it means that the given information is incorrect or the formulas are not being used correctly.

Tips and Tricks

  • Make sure to simplify the equation and solve for 'd' carefully.
  • Use the formula d = (an - a) / (n - 1) or d = (an - a1) / (n - 1) to find the common difference 'd'.
  • Check if the given information is correct by plugging the values into the formula and simplifying the equation.
  • If you get a negative value for 'd', it means that the terms are decreasing, not increasing.

Conclusion

In this article, we answered some frequently asked questions about arithmetic progression (AP) and common difference 'd'. We provided formulas and tips to help you find the common difference 'd' given the first term and the nth term. We also discussed how to simplify the equation and solve for 'd' carefully.