The Sum Of The First 6 Terms Of An Arithmetic Progression Is 18, And The Sixth Term Is Two Times The Third Term. Find The:(i) First Term And The Common Difference.(ii) Sum Of The First 50 Terms.

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Introduction

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. In this article, we will explore the sum of the first 6 terms of an arithmetic progression and use the given information to find the first term and the common difference. We will also calculate the sum of the first 50 terms.

The Sum of the First 6 Terms

Let's denote the first term of the arithmetic progression as a, and the common difference as d. The sum of the first 6 terms can be calculated using the formula:

S6 = 6a + 15d

We are given that the sum of the first 6 terms is 18, so we can write:

6a + 15d = 18

The Sixth Term is Two Times the Third Term

We are also given that the sixth term is two times the third term. The sixth term can be expressed as a + 5d, and the third term can be expressed as a + 2d. Therefore, we can write:

a + 5d = 2(a + 2d)

Simplifying the equation, we get:

a + 5d = 2a + 4d

Subtracting a from both sides, we get:

5d = a + 4d

Subtracting 4d from both sides, we get:

d = a

Finding the First Term and the Common Difference

Now that we have the equation d = a, we can substitute this into the equation 6a + 15d = 18. We get:

6a + 15a = 18

Combining like terms, we get:

21a = 18

Dividing both sides by 21, we get:

a = 18/21

Simplifying the fraction, we get:

a = 6/7

Now that we have the value of a, we can find the value of d. Since d = a, we can substitute a = 6/7 into the equation d = a. We get:

d = 6/7

Finding the Sum of the First 50 Terms

The sum of the first 50 terms can be calculated using the formula:

S50 = 50a + 1245d

Substituting the values of a = 6/7 and d = 6/7, we get:

S50 = 50(6/7) + 1245(6/7)

Simplifying the expression, we get:

S50 = 300/7 + 7458/7

Combining like terms, we get:

S50 = 7758/7

Simplifying the fraction, we get:

S50 = 1106.29

Conclusion

In this article, we have found the first term and the common difference of an arithmetic progression using the given information. We have also calculated the sum of the first 50 terms. The first term is 6/7, and the common difference is also 6/7. The sum of the first 50 terms is approximately 1106.29.

References

Discussion

What do you think about the problem of finding the sum of the first 6 terms of an arithmetic progression? Do you have any questions or comments about the solution? Please share your thoughts in the discussion section below.

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Introduction

In our previous article, we explored the sum of the first 6 terms of an arithmetic progression and used the given information to find the first term and the common difference. We also calculated the sum of the first 50 terms. In this article, we will answer some frequently asked questions about arithmetic progressions.

Q&A

Q: What is an arithmetic progression?

A: 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.

Q: How do I find the first term and the common difference of an arithmetic progression?

A: To find the first term and the common difference, you can use the given information to set up a system of equations. For example, if you know the sum of the first 6 terms and the sixth term is two times the third term, you can use the formulas:

S6 = 6a + 15d

a + 5d = 2(a + 2d)

Solving these equations will give you the values of a and d.

Q: How do I calculate the sum of the first n terms of an arithmetic progression?

A: The sum of the first n terms of an arithmetic progression can be calculated using the formula:

Sn = n/2 (2a + (n-1)d)

Where Sn is the sum of the first n terms, a is the first term, d is the common difference, and n is the number of terms.

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

A: The nth term of an arithmetic progression can be calculated using the formula:

an = a + (n-1)d

Where an is the nth term, a is the first term, d is the common difference, and n is the term number.

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

A: The sum of an infinite arithmetic progression can be calculated using the formula:

S = a / (1 - r)

Where S is the sum, a is the first term, and r is the common ratio.

Q: What is the relationship between the sum of an arithmetic progression and the number of terms?

A: The sum of an arithmetic progression is directly proportional to the number of terms. As the number of terms increases, the sum of the arithmetic progression also increases.

Conclusion

In this article, we have answered some frequently asked questions about arithmetic progressions. We have covered topics such as finding the first term and the common difference, calculating the sum of the first n terms, and finding the sum of an infinite arithmetic progression. We hope that this article has been helpful in clarifying any doubts you may have had about arithmetic progressions.

References

Discussion

Do you have any questions or comments about arithmetic progressions? Please share your thoughts in the discussion section below.