Sequence Terms: Find C_10, C_25, C_200, C_253, C_2k, C_{2k+1}

Hey guys! Let's dive into a cool sequence problem. We've got a sequence $(c_n)$ where the odd terms are always $-1$, and the even terms are always $0$. Our mission, should we choose to accept it (and we do!), is to figure out the first eight terms, and then pinpoint the values of some specific terms like $c_{10}$, $c_{25}$, $c_{200}$, $c_{253}$, and even nail down general formulas for $c_{2k}$ and $c_{2k+1}$. Buckle up; it’s sequence time!

Unpacking the Sequence

First things first, let's break down what this sequence is all about. The problem tells us a super important rule: if the term number ($n$) is odd, the term ($c_n$) is $-1$. If $n$ is even, then $c_n$ is $0$. This is the golden key to solving this puzzle. It's like having a secret code that tells us exactly what each term is. So, before we start listing terms, let’s really make sure we understand this rule. An odd number is any whole number that can’t be divided evenly by 2 (think 1, 3, 5, 7, and so on), while an even number can be (like 2, 4, 6, 8). This simple distinction is what drives the entire sequence.

Listing the First Eight Terms

Alright, let’s get our hands dirty and list out those first eight terms. We'll use our rule like a pro. Remember, odd terms get $-1$, and even terms get $0$. Let's go:

• $c_1$: Since 1 is odd, $c_1 = -1$
• $c_2$: Since 2 is even, $c_2 = 0$
• $c_3$: Since 3 is odd, $c_3 = -1$
• $c_4$: Since 4 is even, $c_4 = 0$
• $c_5$: Since 5 is odd, $c_5 = -1$
• $c_6$: Since 6 is even, $c_6 = 0$
• $c_7$: Since 7 is odd, $c_7 = -1$
• $c_8$: Since 8 is even, $c_8 = 0

So, the first eight terms of our sequence are: $-1, 0, -1, 0, -1, 0, -1, 0$. See the pattern? It’s like a little dance between $-1$ and $0$. Understanding this rhythm is key to cracking the rest of the problem.

Finding Specific Terms: $c_{10}$, $c_{25}$, $c_{200}$, $c_{253}$

Now that we’ve nailed the first eight terms, let’s level up and find some specific ones: $c_{10}$, $c_{25}$, $c_{200}$, and $c_{253}$. The beauty of this sequence is that we don’t need to calculate every term in between – we just need to check if the term number is odd or even. Our golden rule is going to shine here!

Cracking $c_{10}$

Let's start with $c_{10}$. Is 10 odd or even? It’s even! And what do even terms get? A big, fat $0$. So, $c_{10} = 0$. See how easy that was? We didn't have to list out ten terms; we just applied the rule.

Deciphering $c_{25}$

Next up is $c_{25}$. Is 25 odd or even? You guessed it – it’s odd! And odd terms get $-1$. Therefore, $c_{25} = -1$. We're on a roll here, guys!

Unveiling $c_{200}$

Now for a slightly bigger number: $c_{200}$. Is 200 odd or even? It's definitely even. Even terms are $0$, so $c_{200} = 0$. The size of the number doesn't scare us; our rule still works perfectly.

Pinpointing $c_{253}$

Last but not least, let’s tackle $c_{253}$. Is 253 odd or even? It’s odd. So, $c_{253} = -1$. Boom! We've conquered these specific terms. The key takeaway here is that we didn’t need to grind through calculations. We understood the sequence's core rule and applied it strategically.

General Formulas: $c_{2k}$ and $c_{2k+1}$

Okay, we've found specific terms, but let's go even further and find general formulas for $c_{2k}$ and $c_{2k+1}$, where $k$ is any integer. This might sound a bit intimidating, but trust me, it’s super cool and helps us understand the sequence in a much deeper way. Instead of just finding individual terms, we’re going to create formulas that work for any term that fits a certain pattern.

The Magic of $c_{2k}$

Let’s start with $c_{2k}$. What does $2k$ always represent? Well, any number multiplied by 2 is even, right? So, $2k$ will always be an even number, no matter what integer $k$ is. We already know that any even-numbered term in our sequence is $0$. So, we can confidently say that:

$c_{2k} = 0$

That’s it! This formula tells us that any term in the sequence with an even term number will be $0$. We’ve created a general rule that works for an infinite number of terms. How cool is that?

Decoding $c_{2k+1}$

Now let's tackle $c_{2k+1}$. What does $2k+1$ represent? Well, $2k$ is always even, and if we add 1 to an even number, what do we get? An odd number! So, $2k+1$ will always be odd. And we know that any odd-numbered term in our sequence is $-1$. Therefore, we can write:

$c_{2k+1} = -1$

This is another powerful formula. It tells us that any term in the sequence with an odd term number will be $-1$. We now have general formulas for both even and odd terms in our sequence.

Wrapping It Up

So, guys, we’ve really dug deep into this sequence! We started by listing the first eight terms, then we found specific terms like $c_{10}$, $c_{25}$, $c_{200}$, and $c_{253}$. But the real magic happened when we derived the general formulas: $c_{2k} = 0$ and $c_{2k+1} = -1$. These formulas encapsulate the entire pattern of the sequence, allowing us to find any term without having to list them all out. This is the power of understanding patterns and expressing them mathematically. Keep exploring, keep questioning, and keep those mathematical gears turning!