Finding Terms In A Recursive Sequence Calculate A2, A3, And A4

Hey guys! Today, we're diving into the fascinating world of sequences. We've got a sequence defined by a recursive formula, and our mission, should we choose to accept it (and we do!), is to find the values of the first few terms. Specifically, we need to figure out what a₂, a₃, and a₄ are. Don't worry, it's not as intimidating as it sounds. We'll break it down step by step, making sure everyone's on board. So, let's put on our math hats and get started!

The Sequence Definition

First, let's take a close look at the sequence definition. We're given two crucial pieces of information:

• The first term: a₁ = 3
• The recursive formula: aₙ₊₁ = 5/aₙ - 5

The first piece tells us where the sequence starts. It's our anchor point. The recursive formula, on the other hand, tells us how to get from one term to the next. It's like a recipe that uses the previous ingredient (term) to create the next one. This is the heart of the problem, so let's make sure we understand it. The subscript notation (aₙ and aₙ₊₁) might look a bit formal, but it's simply a way of saying "the nth term" and "the next term after the nth term." Think of 'n' as a placeholder for any whole number, like 1, 2, 3, and so on.

Understanding Recursive Formulas

Recursive formulas are super useful in math and computer science because they allow us to define things in terms of themselves. It's like those Russian nesting dolls, where each doll contains a smaller version of itself. In our case, to find a particular term in the sequence, we need to know the term that comes before it. This might seem a bit circular at first, but it works because we have a starting point (a₁). We can use the formula to "unwind" the sequence, one term at a time. Let's illustrate this with an example to make it even clearer. Imagine we want to find a₂. According to the formula, a₂ = 5/a₁ - 5. Since we know a₁ = 3, we can plug it in and calculate a₂. This process of using the previous term to find the next is the essence of recursion.

Now that we have a solid grasp of the sequence definition and recursive formulas, we're ready to roll up our sleeves and actually calculate the values of a₂, a₃, and a₄. Are you guys excited? I know I am!

Calculating a₂

Okay, let's start with finding a₂. This is our first step in unraveling the sequence. To find a₂, we'll use the recursive formula: aₙ₊₁ = 5/aₙ - 5. Remember, 'n' is just a placeholder. In this case, we want to find a₂, which means n+1 = 2. Solving for n, we get n = 1. So, we're looking at the case where n = 1. This means we can substitute n = 1 into our recursive formula: a₁₊₁ = 5/a₁ - 5 which simplifies to a₂ = 5/a₁ - 5. See how we've replaced aₙ₊₁ with a₂ and aₙ with a₁? This is the key to applying the recursive formula. Now comes the fun part – plugging in the value of a₁ that we know. We're given that a₁ = 3. So, we substitute 3 for a₁ in our equation: a₂ = 5/3 - 5. Awesome! We've transformed the problem into a simple arithmetic calculation.

Performing the Calculation

Now, let's do the math. We need to evaluate 5/3 - 5. To do this, we'll first find a common denominator for the two terms. Remember fractions from elementary school? They're coming in handy now! We can rewrite 5 as 15/3. So, our equation becomes: a₂ = 5/3 - 15/3. Now that we have a common denominator, we can subtract the numerators: a₂ = (5 - 15)/3. This simplifies to a₂ = -10/3. And there we have it! We've successfully calculated a₂. It's a fraction, which might seem a bit unexpected, but that's perfectly fine. Sequences can have all sorts of terms, including fractions, decimals, and even more complex numbers. The important thing is that we followed the formula and arrived at a valid result. So, give yourselves a pat on the back! We've conquered the first step. Now, let's move on to finding a₃.

Finding a₃

Alright, we've found a₂, which is -10/3. Now, let's tackle a₃. We'll use the same recursive formula as before: aₙ₊₁ = 5/aₙ - 5. This time, we want to find a₃, so n+1 = 3. Solving for n, we get n = 2. So, we're now in the case where n = 2. This means we can substitute n = 2 into our recursive formula: a₂₊₁ = 5/a₂ - 5 which simplifies to a₃ = 5/a₂ - 5. Notice how the subscript has changed again? This is the beauty of the recursive formula – it adapts to each step of the sequence. Now, the crucial step: plugging in the value of a₂ that we just calculated. We found that a₂ = -10/3. So, we substitute -10/3 for a₂ in our equation: a₃ = 5/(-10/3) - 5. Whoa, we've got a fraction within a fraction! Don't panic! We can handle this.

Dealing with the Complex Fraction

The expression 5/(-10/3) looks a bit intimidating, but it's just a division of fractions. Remember, dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of -10/3 is -3/10. So, we can rewrite 5/(-10/3) as 5 * (-3/10). Now we have a simple multiplication: 5 * (-3/10) = -15/10. We can simplify this fraction by dividing both the numerator and denominator by their greatest common divisor, which is 5. This gives us -3/2. Phew! We've simplified the complex fraction. That's a major accomplishment!. Now we can substitute -3/2 back into our equation for a₃: a₃ = -3/2 - 5. We're almost there!

Completing the Calculation for a₃

We're in the home stretch for finding a₃. We have a₃ = -3/2 - 5. To subtract these terms, we need a common denominator. We can rewrite 5 as 10/2. So, our equation becomes: a₃ = -3/2 - 10/2. Now we can subtract the numerators: a₃ = (-3 - 10)/2. This simplifies to a₃ = -13/2. Boom! We've found a₃! That's another term in our sequence conquered. It was a bit more involved than finding a₂, but we tackled the complex fraction like pros. You guys are doing amazing! We're on a roll. Only one more term to go: a₄.

Time for a₄

We're on the final leg of our journey! We've successfully calculated a₂ and a₃, and now it's time to find a₄. You know the drill by now. We'll use the same recursive formula: aₙ₊₁ = 5/aₙ - 5. This time, we want to find a₄, so n+1 = 4. Solving for n, we get n = 3. We're in the n = 3 case now!. Let's substitute n = 3 into the recursive formula: a₃₊₁ = 5/a₃ - 5 which simplifies to a₄ = 5/a₃ - 5. See how smoothly this recursive formula works? It's like a well-oiled machine!. Now, the familiar step: plugging in the value of a₃ that we calculated earlier. We found that a₃ = -13/2. So, we substitute -13/2 for a₃ in our equation: a₄ = 5/(-13/2) - 5. Another fraction within a fraction! But we're not intimidated anymore, are we? We're fraction-busting experts now!

Conquering the Final Fraction

Let's simplify 5/(-13/2). Remember, dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of -13/2 is -2/13. So, we can rewrite 5/(-13/2) as 5 * (-2/13). This gives us -10/13. Fantastic! We've simplified the complex fraction once again. Now we can substitute -10/13 back into our equation for a₄: a₄ = -10/13 - 5. We're almost at the finish line!

The Grand Finale: Calculating a₄

We're in the final stretch! We have a₄ = -10/13 - 5. To subtract these terms, we need a common denominator. We can rewrite 5 as 65/13. So, our equation becomes: a₄ = -10/13 - 65/13. Now we can subtract the numerators: a₄ = (-10 - 65)/13. This simplifies to a₄ = -75/13. And... we've done it! We've found a₄!. We've successfully calculated the first four terms of the sequence. Give yourselves a huge round of applause! You guys are amazing! We started with a recursive formula and, step by step, unraveled the sequence to find the values we were looking for.

Wrapping Up

So, to recap, we were given a sequence defined by a recursive formula and asked to find a₂, a₃, and a₄. We used the recursive formula to calculate each term, one at a time. Here's what we found:

• a₂ = -10/3
• a₃ = -13/2
• a₄ = -75/13

We navigated fractions within fractions, handled negative numbers, and emerged victorious! This exercise demonstrates the power of recursive formulas and how we can use them to define and explore sequences. I hope you guys enjoyed this journey into the world of sequences! Remember, math is like a puzzle, and with each step we take, we get closer to the solution. Keep practicing, keep exploring, and keep having fun with math!