Arithmetic Progression: Find X, Y, And Z Values

Hey guys! Let's dive into the exciting world of arithmetic progressions and figure out how to solve a classic problem. We're given an arithmetic progression (AP) and our mission, should we choose to accept it, is to find the missing values. So, buckle up and let's get started!

Understanding Arithmetic Progressions

First things first, let's make sure we're all on the same page. An arithmetic progression is a sequence of numbers where the difference between any two consecutive terms is constant. This constant difference is called the common difference, often denoted by 'd'.

Think of it like climbing stairs where each step has the same height. The numbers in the sequence are like the heights you reach as you climb. If you know the height of one step and the height you started at, you can figure out the height of any step!

For example, the sequence 2, 4, 6, 8, 10 is an arithmetic progression with a common difference of 2. Each term is obtained by adding 2 to the previous term. Simple, right?

Key Concepts to Remember:

• First Term (a): The first number in the sequence.
• Common Difference (d): The constant difference between consecutive terms.
• nth Term (an): The term at the nth position in the sequence.

Formula for the nth Term:

The beauty of arithmetic progressions lies in their predictable nature. We have a handy formula to calculate any term in the sequence if we know the first term and the common difference:

an = a + (n - 1) * d

Where:

• an is the nth term
• a is the first term
• n is the position of the term in the sequence
• d is the common difference

This formula is our secret weapon! We'll be using it to crack the problem we're about to tackle. So, keep it in mind.

The Problem at Hand

Okay, let's get down to business. We're given the arithmetic progression:

A = (2, x, 10, y, 18, 22, z, 30)

Our goal is to find the values of x, y, and z. The problem explicitly states that the difference between the terms is constant, which is our key clue that this is indeed an arithmetic progression.

Let's break down what we know:

• The first term (a) is 2.
• We have a sequence of 8 terms.
• We need to find the 2nd (x), 4th (y), and 7th (z) terms.

Before we jump into calculations, let's think about the big picture. We know the first term and we have some other terms in the sequence. If we can figure out the common difference (d), we can use the formula we discussed earlier to find any term we want. It's like connecting the dots – finding 'd' is the key to unlocking the entire sequence.

So, how do we find 'd'? We'll explore that in the next section. Stay tuned!

Finding the Common Difference (d)

Alright, let's put on our detective hats and figure out the common difference (d) in our arithmetic progression. Remember, 'd' is the constant value added to each term to get the next term.

We can use the information we have to our advantage. We know the first term is 2, and we have several other terms in the sequence: 10, 18, 22, and 30. The trick is to use the terms that are farthest apart, as this will give us the most accurate value for 'd'.

Let's focus on the first term (2) and the last term (30). We know that 30 is the 8th term in the sequence. We can use our formula for the nth term:

an = a + (n - 1) * d

Plugging in the values we know:

30 = 2 + (8 - 1) * d

Now, let's simplify and solve for 'd':

30 = 2 + 7d

Subtract 2 from both sides:

28 = 7d

Divide both sides by 7:

d = 4

Eureka! We've found the common difference. It's 4. This means that each term in the sequence is 4 more than the previous term. We're one step closer to finding x, y, and z.

Alternative Approach:

We could have also used any two known terms to find 'd'. For example, we know the 5th term is 18 and the 6th term is 22. The difference between these terms is 22 - 18 = 4, which confirms our value for 'd'.

Now that we have the common difference, we're armed with the key to unlock the missing values. In the next section, we'll use this knowledge to calculate x, y, and z.

Calculating x, y, and z

Okay, guys, this is where the fun really begins! We've successfully found the common difference, d = 4. Now, we can use this to calculate the missing values in our arithmetic progression: x, y, and z.

Let's revisit our arithmetic progression:

A = (2, x, 10, y, 18, 22, z, 30)

Finding x (the 2nd term):

We know that the 2nd term is simply the first term plus the common difference. So:

x = 2 + 4

x = 6

Finding y (the 4th term):

There are a couple of ways we can find y. We could add the common difference to the 3rd term (10), or we could use our formula for the nth term.

Let's use the formula for a bit of practice:

an = a + (n - 1) * d

For the 4th term (y), n = 4. So:

y = 2 + (4 - 1) * 4

y = 2 + 3 * 4

y = 2 + 12

y = 14

Alternatively, we could have simply added the common difference to the 3rd term:

y = 10 + 4

y = 14

Both methods give us the same answer, which is a great way to double-check our work!

Finding z (the 7th term):

Again, we can use either method. Let's stick with the formula for the nth term:

an = a + (n - 1) * d

For the 7th term (z), n = 7. So:

z = 2 + (7 - 1) * 4

z = 2 + 6 * 4

z = 2 + 24

z = 26

And there you have it! We've successfully calculated the values of x, y, and z.

The Solution

Let's recap our findings:

• x = 6
• y = 14
• z = 26

Therefore, the arithmetic progression is:

A = (2, 6, 10, 14, 18, 22, 26, 30)

We've not only solved the problem but also reinforced our understanding of arithmetic progressions. Pat yourselves on the back, guys!

Conclusion

We've successfully navigated the world of arithmetic progressions and found the missing values in our sequence. Remember, the key to solving these problems lies in understanding the concept of the common difference and using the formula for the nth term.

Arithmetic progressions are just one piece of the puzzle in the fascinating world of mathematics. Keep exploring, keep practicing, and you'll be amazed at what you can achieve! Whether you're tackling complex equations or just trying to understand the patterns in the world around you, math is a powerful tool. So, embrace the challenge and keep learning!

I hope this explanation was helpful and clear. If you have any questions or want to explore more math concepts, feel free to ask. Keep up the great work, guys, and I'll see you in the next adventure!