Arithmetic Sequence: Find The First Term If Sum Is 24 & A_6 = 6
Hey guys! Let's dive into a cool math problem today. We're going to figure out the first term of an arithmetic sequence, given that the sum of the terms is 24 and the sixth term is 6. Sounds intriguing, right? We'll use the formula for the nth term, which is a_n = a_1 + (n-1) * r, where a_1 is the first term and r is the common difference. Buckle up, and let's get started!
Understanding Arithmetic Sequences
First things first, let’s make sure we’re all on the same page about what an arithmetic sequence actually is. Think of it like a line of numbers where the gap between each number is the same. This consistent gap is what we call the common difference, often represented by the letter r. So, you start with a number, and to get the next one, you just keep adding the same value. For example, 2, 4, 6, 8... is an arithmetic sequence where the common difference is 2. Easy peasy, right?
The beauty of arithmetic sequences lies in their predictability. Because the difference between terms is constant, we can use formulas to figure out any term in the sequence without having to list them all out. This is super handy, especially when dealing with large sequences. The formula a_n = a_1 + (n-1) * r is our best friend here. It tells us that to find any term (a_n), we just need to know the first term (a_1), the common difference (r), and the position of the term we're looking for (n). Mastering this concept is crucial, as arithmetic sequences pop up in many areas of math and even in real-life situations like calculating simple interest or predicting patterns. So, make sure you've got a good grasp on this before we move on!
Setting Up the Problem: Key Information and Formulas
Okay, so we know we need to find the first term (a_1) of an arithmetic sequence. The problem gives us two key pieces of information: the sum of the terms is 24, and the sixth term (a_6) is 6. We also have the formula for the nth term: a_n = a_1 + (n-1) * r. But, how do we connect the sum of the terms to this formula? That's where another formula comes into play: the sum of the first n terms of an arithmetic sequence, often written as S_n. This formula is given by:
S_n = (n/2) * [2a_1 + (n-1) * r]
This formula might look a bit intimidating at first, but trust me, it's our ticket to solving this problem. It basically tells us that the sum of the first n terms is equal to half the number of terms, multiplied by the sum of twice the first term and the common difference multiplied by (n-1). We need to figure out what 'n' is in our case. The problem states “the sum of the terms is 24”, and since we know a_6, it's reasonable to assume we're talking about the sum of the first six terms. So, n = 6, and S_6 = 24.
Now, let's recap what we know: S_6 = 24, a_6 = 6, n = 6, and the two formulas: a_n = a_1 + (n-1) * r and S_n = (n/2) * [2a_1 + (n-1) * r]. Our goal is to find a_1. We have two equations and two unknowns (a_1 and r), which means we're in business! We can use these equations to create a system and solve for our variables. It's like a mathematical treasure hunt, and we've just found the map and the compass. Let's keep going!
Solving for the Common Difference (r)
Time to put our detective hats on and solve for the common difference, r. Remember, we have two equations: a_n = a_1 + (n-1) * r and S_n = (n/2) * [2a_1 + (n-1) * r]. We also know that a_6 = 6 and S_6 = 24. Let's start by plugging what we know into the first equation:
a_6 = a_1 + (6-1) * r
This simplifies to:
6 = a_1 + 5r (Equation 1)
Now, let's do the same with the sum formula:
S_6 = (6/2) * [2a_1 + (6-1) * r]
Plugging in S_6 = 24, we get:
24 = 3 * [2a_1 + 5r]
Divide both sides by 3:
8 = 2a_1 + 5r (Equation 2)
Now we have a system of two equations:
- 6 = a_1 + 5r
- 8 = 2a_1 + 5r
To solve this, we can use a method called elimination. Notice that the 5r term appears in both equations. Let’s multiply Equation 1 by -2:
-12 = -2a_1 - 10r
Now, add this modified equation to Equation 2:
(-12 + 8) = (-2a_1 + 2a_1) + (-10r + 5r)
This simplifies to:
-4 = -5r
Divide both sides by -5:
r = 4/5
So, we've cracked the code! The common difference, r, is 4/5. That's a big step forward. Now that we know r, we can use it to find a_1, which is exactly what the problem asks for. We're on the home stretch now!
Calculating the First Term (a_1)
Alright, we've found the common difference (r = 4/5). The next step is to plug this value back into one of our equations to solve for the first term, a_1. Let's use Equation 1 because it looks a bit simpler: 6 = a_1 + 5r. Substitute r with 4/5:
6 = a_1 + 5 * (4/5)
Simplify:
6 = a_1 + 4
Now, subtract 4 from both sides:
6 - 4 = a_1
a_1 = 2
Boom! We've got it. The first term of the arithmetic sequence, a_1, is 2. That's the answer we were hunting for! It’s always a good idea to double-check our work to make sure everything adds up. We can plug both a_1 and r back into our original equations to see if they hold true. But for now, let's bask in the glory of solving this problem. We took on a tricky question and came out victorious. Give yourselves a pat on the back, guys!
Final Answer and Verification
So, after all that math magic, we've arrived at our final answer: the first term of the arithmetic sequence (a_1) is 2. Let's quickly verify this to ensure we haven't made any sneaky mistakes along the way.
We know that a_6 = 6 and r = 4/5. We can use the formula a_n = a_1 + (n-1) * r to check if our a_1 value works. Let's plug in n = 6:
a_6 = 2 + (6-1) * (4/5)
a_6 = 2 + 5 * (4/5)
a_6 = 2 + 4
a_6 = 6
Yep, it checks out! The sixth term is indeed 6. Now, let’s verify the sum. We know S_6 should be 24. Using the formula S_n = (n/2) * [2a_1 + (n-1) * r], let's plug in our values:
S_6 = (6/2) * [2 * 2 + (6-1) * (4/5)]
S_6 = 3 * [4 + 5 * (4/5)]
S_6 = 3 * [4 + 4]
S_6 = 3 * 8
S_6 = 24
Awesome! The sum of the first six terms is also 24. This confirms that our solution is correct. We found that the first term (a_1) is 2, and this value satisfies both conditions given in the problem. That feeling when everything clicks into place is just the best, isn't it? You've successfully navigated through an arithmetic sequence problem. Well done!
