Obliczanie Sumy Ciągu Geometrycznego: Krok Po Kroku

Hey guys! Let's dive into a cool math problem. We're going to figure out the sum of the first seven terms of a geometric sequence. This type of problem is super common in math, so understanding it will be a total win. The best part? I'll break it down into easy-to-follow steps. We've got some clues: the sum of the second and fifth terms is 13 1/2, and the sum of the third and sixth terms is 6 3/4. Sounds a bit tricky, right? Don't worry! We'll use these hints to uncover the secrets of the sequence and then calculate our final answer. Ready to get started? Let's do this!

Understanding the Problem and Key Concepts

Alright, before we jump into the calculations, let's make sure we're all on the same page about the basics. 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 'q'. So, if we have a sequence like 2, 4, 8, 16..., the common ratio (q) is 2 because we multiply each number by 2 to get the next one. Also we'll need the formula for the nth term of a geometric sequence: aₙ = a₁ * q^(n-1). Where aₙ is the nth term, a₁ is the first term, q is the common ratio, and n is the term number. Now, we'll be using the formula for the sum of a geometric sequence which will be essential for this problem: Sₙ = a₁ * (1 - qⁿ) / (1 - q). This formula helps us find the sum of the first 'n' terms. It is extremely helpful. Now, let's look at what the problem gives us. We're given two key pieces of information. First: the sum of the second and fifth terms is 13 1/2. Second: the sum of the third and sixth terms is 6 3/4. These are our building blocks. Our goal is to use these clues to find a₁ (the first term), q (the common ratio), and then calculate S₇ (the sum of the first seven terms). Make sense? Great! Let's get to work.

Setting Up Equations from Given Information

Okay, so we know that the sum of the second and fifth terms equals 13 1/2, and the sum of the third and sixth terms equals 6 3/4. Let's translate these statements into mathematical equations using the formula for the nth term. Remember: aₙ = a₁ * q^(n-1). For the first clue (a₂ + a₅ = 13 1/2), we can rewrite this as: a₁q + a₁q⁴ = 27/2. And for the second clue (a₃ + a₆ = 6 3/4), we get: a₁q² + a₁q⁵ = 27/4. Notice that we've expressed each term (a₂, a₅, a₃, and a₆) in terms of a₁, q, and their respective powers. Now, the next step is to simplify these equations to make them easier to solve. A brilliant idea is to factor out the common terms. Let's do that! In the first equation, we can factor out a₁q: a₁q*(1 + q³) = 27/2. In the second equation, factor out a₁q²: a₁q²*(1 + q³) = 27/4. See how this makes the equations more manageable? Now we have two equations that are ready for us to work with. The goal is to find a₁ and q. These are the key ingredients we need to find the sum of the first seven terms, using the sum formula. This is awesome, right? It’s like a puzzle, and each step helps us uncover more clues.

Solving for the Common Ratio (q)

Alright, we're getting closer! We have two equations: 1) a₁q(1 + q³) = 27/2, and 2) a₁q²(1 + q³) = 27/4. Notice something cool? Both equations have the term (1 + q³). This means we can use division to eliminate some variables. Let's divide the second equation by the first equation. This gives us: [a₁q²(1 + q³)] / [a₁q(1 + q³)] = (27/4) / (27/2). Now, the fun part: simplify! The a₁ and (1 + q³) terms cancel out. Also, the 27s cancel out, leaving us with q²/q = (1/2). Simplifying this gives us q = 1/2. Boom! We've found the common ratio! This means that each term in our geometric sequence is half the size of the previous one. This is a big step towards finding the solution. Now that we know 'q', we can plug it back into one of our original equations to solve for a₁. That’s how the math works. We solve one variable, and then use it to solve for another. You could use either equation, but let’s use the first one: a₁q(1 + q³) = 27/2.

Solving for the First Term (a₁) and Sum of Seven Terms

Fantastic! We know q = 1/2. Let’s use this to find a₁. We'll substitute q = 1/2 into our equation a₁q(1 + q³) = 27/2. This gives us: a₁*(1/2)(1 + (1/2)³) = 27/2. Simplify: a₁(1/2)(1 + 1/8) = 27/2. Further simplification: a₁(1/2)(9/8) = 27/2. Then: a₁(9/16) = 27/2. To solve for a₁, multiply both sides by 16/9: a₁ = (27/2)*(16/9). a₁ = 24. Amazing! We've found the first term of our sequence, which is 24! Now that we have both a₁ and q, we can calculate S₇, the sum of the first seven terms. We can use our sum formula Sₙ = a₁ * (1 - qⁿ) / (1 - q). Plugging in the values: S₇ = 24 * (1 - (1/2)⁷) / (1 - 1/2). S₇ = 24 * (1 - 1/128) / (1/2). S₇ = 24 * (127/128) * 2. S₇ = 24 * (127/64). S₇ = 3 * (127/8). S₇ = 37.5. And there you have it! The sum of the first seven terms of the geometric sequence is 37.5. Awesome, right? We broke down the problem step by step, found our common ratio and first term, and then easily calculated the sum. Math is so much fun, isn't it?

Conclusion and Recap

So, what did we learn, guys? We successfully calculated the sum of the first seven terms of a geometric sequence! We started with the given information, converted that into equations, used our algebra skills to find the common ratio (q) and the first term (a₁), and then plugged those values into the sum formula to get our final answer. Isn't it wonderful how each step built upon the previous one? We used equations, division, substitution, and the formula for the sum of a geometric sequence. We learned how to decipher the clues in a math problem, translate them into equations, and solve them using algebraic techniques. That's the beauty of math: it's all connected. If you feel like it, try going through the steps again on your own to reinforce your understanding. And remember, practice makes perfect. The more you work with geometric sequences, the more comfortable and confident you'll become. Keep practicing, stay curious, and keep having fun with math! You got this! This problem is the perfect illustration of how powerful mathematical knowledge can be. You can conquer any problem as long as you remember the core concepts. Keep practicing and good luck!