Finding F(4): A Step-by-Step Function Evaluation

Hey guys! Let's dive into a fun math problem today. We've got a function where we know that $f(1) = 4$, and the function's value at the next step, $f(n+1)$, depends on its current value $f(n)$ with the formula $f(n+1) = -2f(n) + 2$. Our mission, should we choose to accept it (and we do!), is to figure out what $f(4)$ is. No sweat, right? Let's break it down together, step by step, making sure everyone can follow along. We're going to make this super clear and even a little bit stylish, just like Plastik Magazine!

Unpacking the Problem: Understanding the Function

Before we jump into calculations, let's make sure we really get what this function is doing. We are given a recursive function. That just means that to find the value at one point, we need to know the value at the point before it. Think of it like climbing a ladder – you need to be on one rung to get to the next. In our case, we know the value of the first rung, $f(1)$, and we have the rule to get to all the other rungs. The formula $f(n+1) = -2f(n) + 2$ is the key here. It tells us that to get the next value of the function (that's the $f(n+1)$ part), we need to take the current value $f(n)$, multiply it by -2, and then add 2. Simple enough, right? Now, because we want to find $f(4)$, we'll need to find $f(2)$ and $f(3)$ first. We need to climb the ladder one rung at a time! We can apply the given recursive formula $f(n+1) = -2f(n) + 2$ repeatedly, using the result from one step as the input for the next. This step-by-step approach will ensure we arrive at the correct value for $f(4)$, and understanding the mechanics of recursion is a fundamental concept in both mathematics and computer science. So, let's put on our climbing gear and start ascending!

Step 1: Finding f(2)

Okay, time to get our hands dirty with some calculations! We know $f(1) = 4$, and we want to find $f(2)$. Looking at our formula, $f(n+1) = -2f(n) + 2$, we can see that if we set $n = 1$, then $n + 1$ becomes 2. This is exactly what we want! So, we substitute $n = 1$ into the formula: $f(1+1) = -2f(1) + 2$. This simplifies to $f(2) = -2f(1) + 2$. Now we can plug in the value we know for $f(1)$, which is 4: $f(2) = -2(4) + 2$. Let's do the math: $-2$ times 4 is -8, and then we add 2, so $f(2) = -8 + 2 = -6$. Bam! We've found $f(2)$. See? Not so scary. This step highlights the power of substitution and the importance of careful arithmetic. We've taken the given information, plugged it into our formula, and voila, a new piece of the puzzle falls into place. The key here is to treat the formula like a recipe. We have ingredients ($f(1)$ and the formula itself), and by following the recipe (substituting and simplifying), we create a delicious result ($f(2)$). This methodical approach will be crucial as we move forward to find $f(3)$ and ultimately $f(4)$. Keep this momentum going, guys; we are doing great!

Step 2: Finding f(3)

Alright, we're on a roll! We've conquered $f(2)$, and now it's time to set our sights on $f(3)$. We know that $f(2) = -6$, and we're still armed with our trusty formula: $f(n+1) = -2f(n) + 2$. This time, we want to find $f(3)$, so we need to make $n + 1$ equal to 3. What value of $n$ will do the trick? You guessed it, $n = 2$! Let's substitute $n = 2$ into our formula: $f(2+1) = -2f(2) + 2$. This gives us $f(3) = -2f(2) + 2$. Now we plug in the value we just calculated for $f(2)$, which was -6: $f(3) = -2(-6) + 2$. Let's crunch those numbers! -2 times -6 is 12 (remember, a negative times a negative is a positive), and then we add 2, so $f(3) = 12 + 2 = 14$. Boom! Another value down. We're getting closer and closer to our final destination. This step reinforces the concept of building upon previous results. We didn't just magically arrive at $f(3)$; we used our knowledge of $f(2)$ as a stepping stone. This is a common theme in many mathematical problems, and recognizing this pattern can make tackling complex challenges feel much more manageable. We are absolutely nailing it so far!

Step 3: The Grand Finale - Finding f(4)

Okay, guys, this is it! The moment we've been building up to. We're ready to find $f(4)$. We know $f(3) = 14$, and we still have our formula in hand: $f(n+1) = -2f(n) + 2$. This time, we need to make $n + 1$ equal to 4. So, we set $n = 3$. Plugging that into our formula gives us: $f(3+1) = -2f(3) + 2$, which simplifies to $f(4) = -2f(3) + 2$. Now we substitute the value of $f(3)$, which is 14: $f(4) = -2(14) + 2$. Let's do the final calculation: -2 times 14 is -28, and then we add 2, so $f(4) = -28 + 2 = -26$. And there we have it! We've found $f(4)$. This final step is a testament to our systematic approach. We've patiently climbed each rung of the ladder, using our formula and previous results to guide us. This is what problem-solving is all about: breaking down a complex question into smaller, more manageable steps. And the answer we have found is -26.

Conclusion: We Did It!

High fives all around, everyone! We successfully navigated this function evaluation and found that $f(4) = -26$. Wasn't that fun? We took a problem that might have seemed intimidating at first glance and broke it down into easy-to-follow steps. Remember, the key to tackling these kinds of problems is to understand the process, not just memorize formulas. We unpacked the recursive nature of the function, substituted values carefully, and celebrated each small victory along the way. This problem is a great example of how math can be like a puzzle, and we're all puzzle-solving pros now! Keep practicing, keep exploring, and keep that Plastik Magazine style shining through in everything you do. Until next time, stay stylish and keep those brains buzzing!