Finding A₁₀₀ In Arithmetic Progression: A Step-by-Step Guide

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Hey guys! Let's dive into a cool algebra problem today. We're going to figure out how to find a specific term in an arithmetic progression. Arithmetic progressions might sound a bit intimidating, but they're actually super straightforward once you get the hang of them. So, let's break down this problem step by step. We'll make sure to explain everything clearly so that you can tackle similar problems with confidence. Ready? Let's get started!

Understanding Arithmetic Progressions

Before we jump into solving the problem, let's quickly recap what an arithmetic progression actually is. Think of it as a sequence of numbers where the difference between any two consecutive terms is always the same. This constant difference is what we call the common difference.

  • For example, if you start with the number 2 and add 3 each time, you get the sequence 2, 5, 8, 11, and so on. This is an arithmetic progression with a common difference of 3.

  • Understanding this basic concept is crucial because it forms the foundation for solving problems involving arithmetic progressions. We'll be using the properties of arithmetic progressions throughout our solution, so make sure you've got this down!

  • The general form of an arithmetic progression can be written as: a, a + d, a + 2d, a + 3d, ..., where 'a' is the first term and 'd' is the common difference. This form helps us represent any term in the sequence using 'a' and 'd'.

Knowing the general form allows us to express any term in the sequence. For instance, the nth term (aₙ) can be represented as a + (n - 1)d. This formula is a powerful tool for solving problems, as we'll see shortly.

Key Concepts and Formulas

To effectively tackle arithmetic progression problems, there are a couple of key formulas you should have in your toolkit:

  1. The nth term formula: aₙ = a₁ + (n - 1)d. This formula tells you how to find any term in the sequence if you know the first term (a₁) and the common difference (d).
  2. The sum of the first n terms formula: Sₙ = n/2 [2a₁ + (n - 1)d]. This formula helps you calculate the sum of a certain number of terms in the sequence.

For our problem today, we'll primarily be using the nth term formula. It's the key to unlocking the value of a₁₀₀. Make sure you understand how this formula works and how to apply it.

Problem Statement

Now, let's state the problem we're going to solve. We're given an arithmetic progression where:

  • a₂ + a₆ = 44
  • a₅ - a₁ = 20

Our mission, should we choose to accept it (and we do!), is to find the value of a₁₀₀. This means we need to figure out what the 100th term in this sequence is.

This might seem a bit tricky at first, but don't worry! We're going to break it down into manageable steps. The key is to use the information we have to find the first term (a₁) and the common difference (d). Once we have those, we can easily find any term in the sequence, including a₁₀₀.

So, let's roll up our sleeves and get to work!

Setting up the Equations

The first thing we need to do is translate the given information into mathematical equations. Remember the nth term formula we talked about? We're going to use that to express a₂, a₆, a₅, and a₁ in terms of a₁ and d.

  • a₂ can be written as a₁ + d
  • a₆ can be written as a₁ + 5d
  • a₅ can be written as a₁ + 4d

Now, let's substitute these expressions into the equations we were given:

  1. a₂ + a₆ = 44 becomes (a₁ + d) + (a₁ + 5d) = 44
  2. a₅ - a₁ = 20 becomes (a₁ + 4d) - a₁ = 20

See how we've transformed the problem into something more concrete? Now we have two equations with two unknowns (a₁ and d). This means we can solve for them!

Solving for a₁ and d

Let's simplify the equations we set up in the previous step:

  1. (a₁ + d) + (a₁ + 5d) = 44 simplifies to 2a₁ + 6d = 44
  2. (a₁ + 4d) - a₁ = 20 simplifies to 4d = 20

Okay, the second equation is looking pretty straightforward. We can easily solve for d:

  • 4d = 20
  • d = 20 / 4
  • d = 5

Awesome! We've found the common difference, d. Now that we know d, we can plug it back into the first equation to solve for a₁:

  • 2a₁ + 6d = 44
  • 2a₁ + 6(5) = 44
  • 2a₁ + 30 = 44
  • 2a₁ = 44 - 30
  • 2a₁ = 14
  • a₁ = 14 / 2
  • a₁ = 7

Boom! We've found the first term, a₁. So, we now know that a₁ = 7 and d = 5. We're halfway there!

Finding a₁₀₀

Now that we know a₁ and d, finding a₁₀₀ is a piece of cake! We just need to plug these values into the nth term formula:

  • aₙ = a₁ + (n - 1)d

In our case, n = 100, so we have:

  • a₁₀₀ = a₁ + (100 - 1)d
  • a₁₀₀ = 7 + (99) * 5
  • a₁₀₀ = 7 + 495
  • a₁₀₀ = 502

And there you have it! We've found that a₁₀₀ = 502. That means the 100th term in this arithmetic progression is 502.

Conclusion

Alright, guys, we did it! We successfully found a₁₀₀ in the arithmetic progression using the given information. We started by understanding the basic concepts of arithmetic progressions, then we translated the problem into equations, solved for the first term and common difference, and finally, used the nth term formula to find a₁₀₀.

  • Remember, the key to solving these kinds of problems is to break them down into smaller, manageable steps.
  • Don't be afraid to use the formulas and properties of arithmetic progressions.
  • Practice makes perfect, so try solving similar problems to build your confidence.

I hope this step-by-step guide was helpful. If you have any questions, feel free to ask. Keep practicing, and you'll become a pro at solving arithmetic progression problems in no time! Keep up the great work! This whole process is crucial for really nailing down these concepts, so don't skip the practice!