17th Term Of Geometric Sequence: A Step-by-Step Solution

Hey everyone! Today, we're diving into the fascinating world of geometric sequences. Specifically, we're going to tackle a problem where we need to find the 17th term of a sequence, given the first term and the fifth term. Don't worry, it's not as daunting as it sounds! We'll break it down step-by-step, making sure you understand each part of the process. So, let's get started and unlock the secrets of geometric sequences together!

Understanding Geometric Sequences

Before we jump into the problem, let's quickly recap what a geometric sequence actually is. In simple terms, a geometric sequence is a list of numbers where each term is found by multiplying the previous term by a constant value. This constant value is called the common ratio, often denoted by 'r'.

Think of it like this: You start with a number (the first term), and then you consistently multiply by the same number to get the next term, and the next, and so on. This creates a pattern, a sequence, where the ratio between any two consecutive terms is always the same – hence the name "common ratio."

For example, the sequence 2, 6, 18, 54... is a geometric sequence. The first term is 2, and the common ratio is 3 (because 2 * 3 = 6, 6 * 3 = 18, and so on). Understanding this fundamental concept is crucial for solving problems related to geometric sequences, so make sure you've got it down!

To put it more formally, we can express a geometric sequence using a general formula. If we denote the first term as a₁ and the common ratio as r, then the nth term of the sequence, denoted as aₙ, can be calculated using the following formula:

aₙ = a₁ * r^(n-1)

This formula is the key to unlocking many geometric sequence problems, including the one we're about to solve. It tells us that to find any term in the sequence, we just need to know the first term, the common ratio, and the position of the term we're looking for (n). We'll be using this formula extensively in the following sections, so keep it in mind!

Problem Setup: Identifying Key Information

Okay, let's get down to the specific problem we're tackling today. The problem states that we need to identify the 17th term of a geometric sequence. We're given two crucial pieces of information: the first term, a₁, is 16, and the fifth term, a₅, is 150.06. We're also instructed to round both the common ratio and the 17th term to the nearest hundredth. This rounding instruction is important as it tells us the level of precision we need in our final answers.

Before we dive into the calculations, let's pause for a moment and think about what we have and what we need. We know a₁ and a₅, and we want to find a₁₇. The key to bridging this gap is the common ratio, r. If we can figure out the value of r, we can then use the formula we discussed earlier to calculate a₁₇. This is a classic strategy in problem-solving: identify the missing piece of information that connects what you know to what you want to find.

So, our immediate goal is to determine the common ratio, r. How do we do that? We can use the information about a₅ to our advantage. Remember, a₅ is the fifth term in the sequence, which means it's four terms away from the first term, a₁. Each of those four steps involves multiplying by the common ratio, r. This gives us a crucial insight into how we can calculate r. In the next section, we'll put this insight into action and use the geometric sequence formula to find the value of r.

Think of it like climbing stairs. We know the height of the first step (a₁) and the height of the fifth step (a₅). To figure out how much each step rises (the common ratio r), we need to consider the total rise and the number of steps in between. This analogy can help visualize the relationship between the terms in a geometric sequence and how the common ratio connects them.

Calculating the Common Ratio (r)

Now, let's roll up our sleeves and calculate the common ratio, r. This is a crucial step, as it's the key to unlocking the 17th term. We'll use the geometric sequence formula we discussed earlier: aₙ = a₁ * r^(n-1).

We know a₁ = 16 and a₅ = 150.06. We can plug these values into the formula, with n = 5, since we're dealing with the fifth term:

150.06 = 16 * r^(5-1)

This equation relates the fifth term to the first term and the common ratio. Now, it's just a matter of solving for r. First, let's simplify the equation:

150.06 = 16 * r⁴

To isolate r⁴, we need to divide both sides of the equation by 16:

r⁴ = 150.06 / 16 r⁴ = 9.37875

Now, we need to get rid of the exponent. To do this, we'll take the fourth root of both sides of the equation. Remember, the fourth root of a number is the value that, when multiplied by itself four times, equals the original number. You can use a calculator to find the fourth root. Most calculators have a root function, often denoted by a radical symbol with an index (in this case, 4).

r = ⁴√9.37875 r ≈ 1.75

We've found our common ratio! It's approximately 1.75. Remember, the problem asked us to round to the nearest hundredth, so we've adhered to that instruction. This value of r tells us that each term in the sequence is roughly 1.75 times larger than the previous term. This is a significant piece of information, as it allows us to move forward and calculate the 17th term.

It's worth noting that when dealing with even roots (like the fourth root), there could theoretically be both positive and negative solutions. However, in the context of geometric sequences, we typically focus on the positive root unless the problem specifies otherwise. In this case, since the terms are increasing, it makes sense to consider the positive root as the common ratio.

Calculating the 17th Term (a₁₇)

Great job, guys! We've successfully calculated the common ratio, r, which is approximately 1.75. Now, we're just one step away from our final answer: finding the 17th term, a₁₇. We'll use the same geometric sequence formula we used before: aₙ = a₁ * r^(n-1).

This time, we're looking for a₁₇, so n = 17. We know a₁ = 16 and r ≈ 1.75. Let's plug these values into the formula:

a₁₇ = 16 * (1.75)^(17-1) a₁₇ = 16 * (1.75)¹⁶

Now, it's time to calculate (1.75)¹⁶. This is where a calculator comes in handy, especially one with an exponent function (often denoted by a '^' symbol or a 'yˣ' button). Calculating 1.75 raised to the power of 16 will give us a large number.

(1.75)¹⁶ ≈ 14198.57

Now, we multiply this result by the first term, 16:

a₁₇ ≈ 16 * 14198.57 a₁₇ ≈ 227177.12

We've found the 17th term! It's approximately 227177.12. Again, the problem instructed us to round to the nearest hundredth, and we've done so. This is a large number, which makes sense given that the common ratio is greater than 1, meaning the sequence is growing exponentially. So, the 17th term in the sequence is significantly larger than the first term.

This calculation demonstrates the power of geometric sequences. Even with a relatively small common ratio, the terms can grow very quickly as you move further down the sequence. This principle is applied in various real-world scenarios, such as compound interest, population growth, and radioactive decay.

Final Answer and Recap

Alright, we've reached the finish line! We've successfully identified the 17th term of the geometric sequence. Let's recap our findings and state the final answer.

The problem asked us to find the 17th term of a geometric sequence where a₁ = 16 and a₅ = 150.06. We were also instructed to round the common ratio and the 17th term to the nearest hundredth.

First, we calculated the common ratio, r, using the formula aₙ = a₁ * r^(n-1). We found that:

r ≈ 1.75

Then, we used this value of r to calculate the 17th term, a₁₇, again using the same formula. We found that:

a₁₇ ≈ 227177.12

Therefore, the 17th term of the geometric sequence, rounded to the nearest hundredth, is 227177.12. Woohoo! We did it!

Let's quickly recap the steps we took to solve this problem:

1. Understood the concept of geometric sequences: We refreshed our understanding of what geometric sequences are and the role of the common ratio.
2. Set up the problem: We identified the given information (a₁ and a₅) and what we needed to find (a₁₇).
3. Calculated the common ratio (r): We used the geometric sequence formula to solve for r.
4. Calculated the 17th term (a₁₇): We used the calculated value of r and the formula to find a₁₇.
5. Stated the final answer: We clearly presented our solution, rounded to the nearest hundredth as instructed.

By breaking down the problem into smaller, manageable steps, we were able to solve it effectively. This approach is applicable to many mathematical problems and can help you tackle even the most challenging ones. Remember, practice makes perfect, so keep exploring and solving geometric sequence problems!

I hope this step-by-step guide has been helpful in understanding how to find the 17th term of a geometric sequence. Geometric sequences can seem intimidating at first, but with a solid understanding of the formula and a systematic approach, you can conquer them with confidence. Keep practicing, keep exploring, and most importantly, keep having fun with math! Until next time, peace out! Guys, you are amazing!