Sequence Solver: Finding Terms In 4, 8, 12...

by Rajiv Sharma 46 views

Hey guys! Let's dive into this math problem together and figure out how to find the indicated term in the sequence. It looks like we've got a pretty straightforward arithmetic sequence here, and we're going to break it down step by step so you can not only solve this problem but also understand the underlying concepts. So, grab your thinking caps, and let’s get started!

Understanding Arithmetic Sequences

Before we jump into solving the specific problem, let's make sure we're all on the same page about arithmetic sequences. An arithmetic sequence is simply a list of numbers where the difference between any two consecutive terms is constant. This constant difference is called the common difference. Recognizing this pattern is key to cracking these types of problems. In our case, we have the sequence 4, 8, 12, 16, 20... Can you spot the common difference?

It's pretty clear, right? We're adding 4 each time to get the next term. This means our common difference (often denoted as 'd') is 4. Understanding the common difference is like having the secret code to unlock the sequence. Now that we've identified the common difference, let's talk about how we can use it to find any term in the sequence. There's a handy formula that makes this super easy.

The formula for the nth term (often written as an) of an arithmetic sequence is:

an = a1 + (n - 1)d

Where:

  • an is the nth term we want to find
  • a1 is the first term in the sequence
  • n is the position of the term we want to find (e.g., 72nd term, 80th term, etc.)
  • d is the common difference

This formula is your best friend when dealing with arithmetic sequences. It allows you to jump directly to any term without having to list out all the terms in between. Imagine trying to find the 100th term by just adding 4 repeatedly – that would take forever! This formula is our shortcut.

Let's break down why this formula works. The term (n - 1)d represents the total difference added to the first term to reach the nth term. We subtract 1 from n because we start counting the difference from the second term (the first term doesn't have a 'difference' added to it). By adding this total difference to the first term (a1), we land exactly on the nth term (an). Make sense? Awesome!

Now that we understand the formula and the concept of arithmetic sequences, we're ready to tackle the specific problem. We’ll plug in the values we know and solve for the term we're trying to find. Let's get into it!

Applying the Formula to Find the Term

Okay, so now that we've got a handle on the arithmetic sequence formula, let's put it to work! In our problem, the sequence is 4, 8, 12, 16, 20..., and we need to figure out a specific term. We know the first term (a1) is 4, and the common difference (d) is also 4. Remember, we found the common difference by observing that each term increases by 4. Now, let's look at the options given to us:

a) 72 b) 80 c) 88 d) 76 e) 84

These options represent possible terms in the sequence. Our goal is to figure out which of these terms corresponds to an integer value of 'n' in our formula. In other words, we want to find which of these numbers fits the pattern of the sequence. To do this, we'll rearrange our formula to solve for 'n'.

Starting with our formula: an = a1 + (n - 1)d

We'll rearrange it to isolate 'n':

  1. Subtract a1 from both sides: an - a1 = (n - 1)d
  2. Divide both sides by d: (an - a1) / d = n - 1
  3. Add 1 to both sides: (an - a1) / d + 1 = n

So our rearranged formula to find 'n' is:

n = (an - a1) / d + 1

This formula tells us the position of a term (an) in the sequence. If 'n' turns out to be a whole number (an integer), then an is indeed a term in the sequence. If 'n' is not an integer, then an is not part of the sequence. Now we're ready to test each of our options!

We'll plug in each of the given values (72, 80, 88, 76, and 84) for an in our formula and see which one gives us a whole number for 'n'. Remember, a1 is 4 and d is 4. Let's get calculating!

Calculating 'n' for Each Option

Alright, time to crunch some numbers! We're going to use our rearranged formula, n = (an - a1) / d + 1, and plug in each of the possible terms to see which one gives us a whole number for 'n'. This will tell us which of the options is actually a term in the sequence.

Let's start with option a) 72:

n = (72 - 4) / 4 + 1 n = 68 / 4 + 1 n = 17 + 1 n = 18

So, for 72, we get n = 18. This is a whole number! That means 72 is the 18th term in the sequence. Exciting! Let's keep going and check the other options. This is a crucial step to confirm we have the correct solution and to understand why the other options might not fit.

Now, let's try option b) 80:

n = (80 - 4) / 4 + 1 n = 76 / 4 + 1 n = 19 + 1 n = 20

For 80, we get n = 20. Another whole number! So, 80 is the 20th term in the sequence. It seems like we're on the right track. But we still need to check all the options to be absolutely sure we've identified the correct one in the context of the original problem.

Next up, option c) 88:

n = (88 - 4) / 4 + 1 n = 84 / 4 + 1 n = 21 + 1 n = 22

For 88, we get n = 22. Yet another whole number! 88 is the 22nd term. This is interesting – we have three options that work so far. This highlights the importance of checking all options and understanding the specific question being asked. We need to think about what the question is actually asking us to find.

Moving on to option d) 76:

n = (76 - 4) / 4 + 1 n = 72 / 4 + 1 n = 18 + 1 n = 19

For 76, we get n = 19. This is also a whole number, meaning 76 is the 19th term in the sequence. Now we have four possible answers! This might seem confusing, but it’s a good reminder to always double-check our work and the original question.

Finally, let's check option e) 84:

n = (84 - 4) / 4 + 1 n = 80 / 4 + 1 n = 20 + 1 n = 21

For 84, we get n = 21. This is, again, a whole number, making 84 the 21st term in the sequence. Wow! All five options turned out to be terms in the sequence. This means we need to go back to the original question and make sure we understand what it's asking for. There might be a specific term it's looking for, or perhaps there's an error in the question itself. Let's revisit the original problem statement.

Re-evaluating the Question and Identifying the Correct Answer

Okay, guys, this is a great learning moment! We’ve done all the math correctly, and we’ve found that all the given options (72, 80, 88, 76, and 84) are indeed terms in the arithmetic sequence 4, 8, 12, 16, 20... But this means we need to take a closer look at the original question. Sometimes, in math problems, the wording can be a little tricky, or there might be an implied condition that we need to consider.

The original question states: "En cada caso, calcula el término indicado. 4. 18 4; 8; 12; 16; 20; ..24 a)72 d)76 b) 80 e) 84 c) 88"

After reviewing, it seems like there might be a slight misunderstanding or missing information in the original question. The phrase "calcula el término indicado" suggests that there should be a specific term indicated, but it's not explicitly stated which term we're supposed to calculate. The numbers 4. 18 and 24 seem out of place and might be remnants of a larger problem or a typo.

Given the options and the sequence, we can infer that the question likely intended to ask if these numbers are terms in the sequence. We've already confirmed that they all are. However, without a specific term indicated, we can't definitively choose one answer over the others.

In a real-world scenario, this is where you might ask for clarification from your teacher or instructor. It's always better to be sure you understand the question before providing an answer. For the purpose of this exercise, we've demonstrated how to determine if a number is a term in an arithmetic sequence. We've used the formula, rearranged it to solve for 'n', and checked each option. This is the core skill the problem likely intended to assess.

So, while we can't pinpoint a single