Finding The Term Number Of 66 In An Arithmetic Sequence

Hey guys! Let's dive into a cool math problem today, specifically about arithmetic sequences. We've got a sequence where the sum of the 9th and 4th terms is 78, and the difference between the 10th and 5th terms is 30. The big question is: where does the number 66 fit into this sequence? Which term is it?

Understanding Arithmetic Sequences

Before we jump into solving this, let's quickly refresh what an arithmetic sequence is. Think of it as a list of numbers where the gap between each number is the same. This consistent gap is called the "common difference." For example, 2, 4, 6, 8... is an arithmetic sequence where the common difference is 2. Each term in the sequence can be represented using a formula, which we'll use shortly.

In arithmetic sequences, identifying patterns is key. We need to figure out the first term and the common difference to unravel the mystery of where 66 sits. The beauty of arithmetic sequences is their predictable nature. Each term is simply the previous term plus the common difference. This predictable pattern allows us to express any term in the sequence using a simple formula. So, by understanding the relationships between the terms, we can unlock the secrets of the sequence and find the position of any number within it. It's like having a treasure map where each term guides us closer to the final destination. In this case, our treasure is the position of 66, and the map is the information provided in the problem.

Setting Up the Equations

Now, let's translate the given information into mathematical equations. This is a crucial step because it transforms the word problem into a language we can easily work with. Remember, Uₙ represents the nth term of the sequence. We're told that U₉ + U₄ = 78 and U₁₀ - U₅ = 30. These are our two golden clues. To use these clues effectively, we need to express each term (U₉, U₄, U₁₀, and U₅) in terms of the first term (let's call it 'a') and the common difference (let's call it 'b').

Here's where the formula for the nth term of an arithmetic sequence comes in handy: Uₙ = a + (n - 1)b. So, U₉ = a + 8b, U₄ = a + 3b, U₁₀ = a + 9b, and U₅ = a + 4b. Now we can rewrite our given equations using these expressions. Replacing the terms with their formulas, we get:

• (a + 8b) + (a + 3b) = 78
• (a + 9b) - (a + 4b) = 30

See how we've transformed the problem into a system of equations? This is a standard technique in algebra, and it allows us to solve for the unknowns, 'a' and 'b'. By carefully substituting the formulas for each term, we've created a clear pathway to find the values we need. It's like building a bridge from the given information to the solution.

Solving for 'a' and 'b'

Let's simplify these equations and solve for 'a' and 'b'. Combining like terms in the first equation, we get 2a + 11b = 78. In the second equation, the 'a' terms cancel out, leaving us with 5b = 30. This is fantastic news because we can immediately solve for 'b'! Dividing both sides of 5b = 30 by 5, we find that b = 6. So, the common difference is 6. That's one piece of the puzzle solved!

Now that we know 'b', we can substitute it back into the first equation (2a + 11b = 78) to find 'a'. Replacing 'b' with 6, we get 2a + 11(6) = 78, which simplifies to 2a + 66 = 78. Subtracting 66 from both sides gives us 2a = 12, and dividing by 2, we find that a = 6. So, the first term of the sequence is also 6. We've now successfully unlocked both 'a' and 'b', which are the keys to understanding this arithmetic sequence. With these values in hand, we're ready to tackle the main question: where does 66 fit in?

Finding the Position of 66

Okay, we know the first term (a = 6) and the common difference (b = 6). Our mission now is to find which term in the sequence equals 66. In other words, we need to find the value of 'n' such that Uₙ = 66. Let's use the formula for the nth term again: Uₙ = a + (n - 1)b. We'll substitute the values we know (Uₙ = 66, a = 6, and b = 6) into the formula and solve for 'n'.

Plugging in the values, we get 66 = 6 + (n - 1)6. Now it's just a matter of simplifying and solving for 'n'. First, subtract 6 from both sides: 60 = (n - 1)6. Then, divide both sides by 6: 10 = n - 1. Finally, add 1 to both sides: n = 11. So, 66 is the 11th term in the sequence! We've cracked the code and found the position of 66. It's like finding the hidden treasure at the end of a challenging quest.

Conclusion

To recap, we were given information about an arithmetic sequence and asked to find which term the number 66 belonged to. We started by understanding the properties of arithmetic sequences and setting up equations based on the given information. Then, we solved for the first term ('a') and the common difference ('b'). Finally, we used these values to find the position of 66 in the sequence. And there you have it, guys! We successfully navigated through this arithmetic sequence problem and discovered that 66 is the 11th term. Keep practicing, and you'll become arithmetic sequence masters in no time!