The Fourth Term And The Sum Of An Arithmetic Progression Are 4 And 5, Respectively. Find:(i) The First Term And The Common Difference.(ii) The Fortieth Term Of The Progression.

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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 properties of an arithmetic progression and use the given information to find the first term and the common difference, as well as the fortieth term of the progression.

Given Information

Let's denote the first term of the arithmetic progression as 'a' and the common difference as 'd'. We are given that the fourth term of the progression is 4, and the sum of the first 40 terms is 5.

Equations

We can write the following equations based on the given information:

  • The fourth term of the progression is 4: a + 3d = 4
  • The sum of the first 40 terms is 5: 40/2 [2a + (40-1)d] = 5

Simplifying the Equations

Let's simplify the second equation:

40/2 [2a + (40-1)d] = 5 20 [2a + 39d] = 5 2a + 39d = 5/20 2a + 39d = 1/4

Now we have two equations:

a + 3d = 4 2a + 39d = 1/4

Solving the Equations

We can solve these equations using the method of substitution or elimination. Let's use the elimination method.

Multiply the first equation by 2 to get:

2a + 6d = 8

Now subtract the second equation from this new equation:

(2a + 6d) - (2a + 39d) = 8 - 1/4 -33d = 31/4

Now divide by -33:

d = -31/4 * -1/33 d = 31/132

Now that we have the value of d, we can substitute it into one of the original equations to find the value of a. Let's use the first equation:

a + 3d = 4 a + 3(31/132) = 4 a + 31/44 = 4 a = 4 - 31/44 a = (176 - 31)/44 a = 145/44

The First Term and the Common Difference

We have found the values of a and d:

a = 145/44 d = 31/132

The Fortieth Term of the Progression

Now that we have the values of a and d, we can find the fortieth term of the progression using the formula:

an = a + (n-1)d

where an is the nth term of the progression.

For the fortieth term, n = 40:

a40 = a + (40-1)d a40 = 145/44 + 39(31/132) a40 = 145/44 + 1211/132 a40 = (145*3 + 1211)/132 a40 = (435 + 1211)/132 a40 = 1646/132 a40 = 823/66

Conclusion

In this article, we used the given information to find the first term and the common difference of an arithmetic progression, as well as the fortieth term of the progression. We simplified the equations, solved them using the elimination method, and found the values of a and d. We then used these values to find the fortieth term of the progression.

References

  • [1] "Arithmetic Progression" by Khan Academy
  • [2] "Arithmetic Progression" by Math Open Reference

Further Reading

  • "Arithmetic Progression" by Wikipedia
  • "Arithmetic Progression" by Wolfram MathWorld
    The Fourth Term and the Sum of an Arithmetic Progression: Q&A ===========================================================

Introduction

In our previous article, we explored the properties of an arithmetic progression and used the given information to find the first term and the common difference, as well as the fortieth term of the progression. In this article, we will answer some common questions related to arithmetic progressions and provide additional examples to help solidify your understanding.

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 following formula:

an = a + (n-1)d

where an is the nth term of the progression, a is the first term, n is the term number, and d is the common difference.

You can also use the given information to set up equations and solve for a and d.

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

A: To find the sum of the first n terms of an arithmetic progression, you can use the following formula:

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

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

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

A: The formula for the nth term of an arithmetic progression is:

an = a + (n-1)d

where an is the nth term of the progression, a is the first term, n is the term number, and d is the common difference.

Q: How do I find the fortieth term of an arithmetic progression?

A: To find the fortieth term of an arithmetic progression, you can use the formula:

a40 = a + (40-1)d

where a40 is the fortieth term, a is the first term, and d is the common difference.

Q: What is the relationship between the first term, the common difference, and the sum of the first n terms of an arithmetic progression?

A: The first term, the common difference, and the sum of the first n terms of an arithmetic progression are related by the following formula:

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

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

Q: Can I use the formula for the sum of the first n terms to find the first term and the common difference?

A: Yes, you can use the formula for the sum of the first n terms to find the first term and the common difference. However, you will need to solve a system of equations to find the values of a and d.

Examples

Example 1: Find the first term and the common difference of an arithmetic progression given the sum of the first 10 terms.

Let's say the sum of the first 10 terms is 50. We can use the formula for the sum of the first n terms to set up an equation:

10/2 [2a + (10-1)d] = 50

Simplifying the equation, we get:

5 [2a + 9d] = 50

Expanding the equation, we get:

10a + 45d = 50

Now we can solve for a and d using the elimination method.

Example 2: Find the fortieth term of an arithmetic progression given the first term and the common difference.

Let's say the first term is 2 and the common difference is 3. We can use the formula for the nth term to find the fortieth term:

a40 = a + (40-1)d a40 = 2 + (40-1)3 a40 = 2 + 39(3) a40 = 2 + 117 a40 = 119

Conclusion

In this article, we answered some common questions related to arithmetic progressions and provided additional examples to help solidify your understanding. We also explored the relationship between the first term, the common difference, and the sum of the first n terms of an arithmetic progression.

References

  • [1] "Arithmetic Progression" by Khan Academy
  • [2] "Arithmetic Progression" by Math Open Reference

Further Reading

  • "Arithmetic Progression" by Wikipedia
  • "Arithmetic Progression" by Wolfram MathWorld