Increasing Sequence Function Type: Initial Value Analysis

Hey guys! Let's dive into the fascinating world of sequences and functions, specifically focusing on increasing sequences and how we can determine the type of function that represents them. Today, we're tackling a scenario where we have an increasing sequence with a given percent rate of change and a specific term value. Our goal? To figure out which type of function, in terms of its initial value, models this sequence. So, buckle up and let's get started!

Decoding the Problem: Rate of Change and the Sixth Term

So, we've got this increasing sequence, right? And it's growing at a rate of 3.58%. That's our key piece of information about how the sequence is changing. In the context of sequences, this percentage rate of change is super important because it tells us that each term is a certain percentage larger than the one before it. This is a hallmark of exponential growth, which we'll get into more later.

Now, we also know that the sixth term in this sequence clocks in at 49.399. This is like a specific landmark on our sequence's journey, giving us a fixed point to work with. Think of it as a coordinate on a graph – it tells us exactly where the sequence is at a particular step. With this info, we can work backward or forward to figure out other terms and, ultimately, the function that governs the whole sequence.

Here's why these two pieces of information are so crucial. The rate of change (3.58%) hints strongly at the type of function we're dealing with – exponential, most likely. The sixth term (49.399) gives us a concrete value, which acts as an anchor point. We can use these together to nail down the function's specific parameters, like its initial value. It's like having a map (the rate of change) and a landmark (the sixth term) – we can use them to pinpoint our location (the function).

To truly grasp this, imagine the sequence as a series of stepping stones, each one a little higher than the last. The 3.58% rate of change is the consistent incline of the path, and the sixth stepping stone being at 49.399 is like knowing the height at a specific point along that path. We need to use both the incline and the height at that point to figure out where the path started – the initial value.

Identifying the Function Type: Exponential Functions in the Spotlight

When we talk about a sequence with a consistent percent rate of change, we're almost always talking about an exponential function. Why? Because exponential functions are defined by their multiplicative growth. Instead of adding the same amount each time (like in an arithmetic sequence), each term is multiplied by a constant factor. That factor is directly related to our percent rate of change.

Think of it this way: a 3.58% rate of change means that each term is 103.58% of the previous term. We get that 103.58% by adding the 3.58% increase to the original 100% of the previous term. Mathematically, we represent this as a decimal by dividing by 100, which gives us 1.0358. This number, 1.0358, is our growth factor, the magic multiplier that defines our exponential function.

Now, let's chat about the general form of an exponential function. It usually looks something like this: h(x) = a(b)^x, where:

• h(x) is the value of the function at a given input x
• a is the initial value (the value when x = 0)
• b is the growth factor (the number we multiply by each time)
• x is the input variable (often representing the term number in a sequence)

This form is super versatile. It models tons of real-world phenomena, from population growth to compound interest. The key thing to remember is that the base, b, is what determines the growth rate. If b is greater than 1, we have exponential growth. If b is between 0 and 1, we have exponential decay.

In our specific problem, the 3.58% rate of change translates directly into a growth factor of 1.0358. This immediately tells us that our function will have the form h(x) = a(1.0358)^x. The only thing left to figure out is the initial value, a. That's where the sixth term comes in handy. We'll use it as a kind of decoder ring to unlock the value of a.

Calculating the Initial Value: Working Backwards

Okay, so we know our function looks like h(x) = a(1.0358)^x, and we know that the sixth term, h(5) (remember, in sequences we often start counting terms from 0), is 49.399. Notice that I said h(5) and not h(6) here. This is because the 'initial value' is considered to be at x=0, therefore the sixth term will be the fifth increase in the sequence. Now, we can plug these values into our equation and solve for a, the initial value. It's like solving a puzzle, and we have all the pieces we need!

Here's how the math works:

1. Substitute: We know h(5) = 49.399 and x = 5, so we get: 49.399 = a(1.0358)^5
2. Calculate the growth factor raised to the power: Calculate 1.0358 raised to the power of 5, which is approximately 1.19399. So our equation now looks like this: 49.399 = a * 1.19399
3. Isolate 'a': To get 'a' by itself, we divide both sides of the equation by 1.19399: a = 49.399 / 1.19399
4. Solve for 'a': Performing the division, we find that a is approximately 41.3725.

So, our initial value, a, is around 41.3725. This means that the first term in our sequence (when x = 0) was roughly 41.3725. This is a critical piece of information because it tells us where the sequence started before it began its 3.58% climb with each subsequent term.

This process of working backwards from a known term to find the initial value is a common technique in dealing with sequences and functions. It's like tracing a path back to its starting point, which gives us a complete picture of the sequence's behavior.

The Final Function: Putting It All Together

Alright, we've done the detective work, cracked the code, and now it's time to reveal the function that represents our increasing sequence. We know it's an exponential function, we know the growth factor is 1.0358 (from the 3.58% rate of change), and we've calculated the initial value to be approximately 41.3725. So, we can now confidently write out the full function!

Remember the general form of an exponential function: h(x) = a(b)^x.

We've found:

• a (the initial value) ≈ 41.3725
• b (the growth factor) = 1.0358

Plugging these values into our general form, we get:

h(x) = 41.3725(1.0358)^x

This is it! This function perfectly models our increasing sequence. It tells us exactly how the sequence grows, starting from its initial value of approximately 41.3725 and increasing by 3.58% with each term. It's like having the blueprint for the entire sequence, allowing us to calculate any term we want.

Now, let's take a moment to reflect on what we've accomplished. We started with a description of a sequence and a single term value. By understanding the properties of exponential functions and using a little bit of algebra, we've constructed a complete mathematical model. This is a powerful skill that can be applied to all sorts of problems involving growth and change.

So, the final answer is that the function created from the sequence in terms of the initial value is an exponential function of the form h(x) = 41.3725(1.0358)^x. We successfully navigated the problem by identifying the function type, utilizing the given rate of change and term value, and solving for the initial value. Great job, guys!