Math Problems: Solving Series And Sums

Hey guys! Let's dive into some cool math problems involving series and sums. We've got four interesting challenges here that will help us brush up on our arithmetic skills. Let's break them down one by one and make sure we understand each step. So grab your pencils, and let’s get started!

a) Solve: (12 + 24 + 36 + 444) : (6 + 12 + 18 ... + 222)

Okay, so our first task involves solving a division problem with sums. The expression is (12 + 24 + 36 + 444) : (6 + 12 + 18 ... + 222). This looks a bit intimidating at first, but don't worry, we'll tackle it together.

First, let’s simplify the numerator. We need to add these numbers up: 12 + 24 + 36 + 444. If we do the addition carefully,

12 + 24 = 36

36 + 36 = 72

72 + 444 = 516

So, the sum of the numerator is 516. Great! Now let’s move on to the denominator.

The denominator is a series: 6 + 12 + 18 ... + 222. Notice that this is an arithmetic progression, where each term increases by 6. To find the sum of an arithmetic series, we can use the formula:

Sum = (n / 2) * (first term + last term)

Where n is the number of terms in the series. We know the first term is 6 and the last term is 222. But how do we find n? To find the number of terms, we can use the formula for the nth term of an arithmetic sequence:

an = a1 + (n - 1) * d

Here, an is the nth term (222), a1 is the first term (6), and d is the common difference (6). Let’s plug in the values:

222 = 6 + (n - 1) * 6

Now, let's solve for n:

222 = 6 + 6n - 6

222 = 6n

n = 222 / 6

n = 37

So, there are 37 terms in the series. Now we can find the sum of the denominator using the sum formula:

Sum = (37 / 2) * (6 + 222)

Sum = (37 / 2) * 228

Sum = 37 * 114

Sum = 4218

Awesome! We’ve calculated the sum of the denominator to be 4218. Now we can go back to our original expression and perform the division:

516 : 4218

This simplifies to:

516 / 4218 = 258 / 2109 = 86 / 703

So, the final answer for part a) is 86 / 703. Great job, guys! We tackled that one step by step and got through it. Remember, breaking down complex problems into smaller parts makes them much easier to handle.

b) Calculate: (4 + 8 + 12 ... + 2012) : (2 + 4 + 6 ... + 1006)

Next up, we need to calculate the value of another division problem with arithmetic series: (4 + 8 + 12 ... + 2012) : (2 + 4 + 6 ... + 1006). This looks similar to the first problem, but with different numbers, so we'll use the same approach. Let’s handle the numerator and the denominator separately.

First, let's look at the numerator: 4 + 8 + 12 ... + 2012. This is an arithmetic series with a common difference of 4. The first term is 4, and the last term is 2012. We need to find the number of terms, n, to use our sum formula. We'll use the same formula as before:

an = a1 + (n - 1) * d

Here, an is 2012, a1 is 4, and d is 4. Plugging in the values:

2012 = 4 + (n - 1) * 4

Let's solve for n:

2012 = 4 + 4n - 4

2012 = 4n

n = 2012 / 4

n = 503

So, there are 503 terms in the numerator series. Now we can find the sum using the formula:

Sum = (n / 2) * (first term + last term)

Sum = (503 / 2) * (4 + 2012)

Sum = (503 / 2) * 2016

Sum = 503 * 1008

Sum = 507024

Fantastic! The sum of the numerator is 507024. Now, let's move on to the denominator: 2 + 4 + 6 ... + 1006. This is another arithmetic series, but this time the common difference is 2. The first term is 2, and the last term is 1006. Let's find the number of terms, n:

1006 = 2 + (n - 1) * 2

Solving for n:

1006 = 2 + 2n - 2

1006 = 2n

n = 1006 / 2

n = 503

There are 503 terms in the denominator series. Now we find the sum:

Sum = (503 / 2) * (2 + 1006)

Sum = (503 / 2) * 1008

Sum = 503 * 504

Sum = 253512

Great! The sum of the denominator is 253512. Now we divide the numerator sum by the denominator sum:

507024 / 253512 = 2

So, the answer for part b) is 2. You guys are doing an amazing job keeping up! We're halfway there. Let’s keep pushing through.

c) Determine: 6 + 11 + 16 + 21 + ... + 506

Now, let’s figure out the sum of this arithmetic series: 6 + 11 + 16 + 21 + ... + 506. We need to find the sum of this series. First, we identify that this is an arithmetic sequence where the common difference is 5. The first term is 6, and the last term is 506.

We need to find the number of terms, n. Let’s use the formula:

an = a1 + (n - 1) * d

Here, an is 506, a1 is 6, and d is 5. Plugging in the values:

506 = 6 + (n - 1) * 5

Solving for n:

506 = 6 + 5n - 5

506 = 1 + 5n

505 = 5n

n = 505 / 5

n = 101

So, there are 101 terms in the series. Now we can find the sum using the formula:

Sum = (n / 2) * (first term + last term)

Sum = (101 / 2) * (6 + 506)

Sum = (101 / 2) * 512

Sum = 101 * 256

Sum = 25856

Therefore, the sum of the series is 25856. Wonderful! We’re on a roll. Let’s move on to the last problem.

d) Calculate: 4 + 9 + 14 + 19 + ... + 1004

Alright, our final challenge is to calculate another arithmetic series: 4 + 9 + 14 + 19 + ... + 1004. Just like the previous problem, we need to find the sum of this series. This is an arithmetic sequence with a common difference of 5. The first term is 4, and the last term is 1004.

Let's find the number of terms, n, using the formula:

an = a1 + (n - 1) * d

Here, an is 1004, a1 is 4, and d is 5. Let’s plug in the values:

1004 = 4 + (n - 1) * 5

Now, let's solve for n:

1004 = 4 + 5n - 5

1004 = 5n - 1

1005 = 5n

n = 1005 / 5

n = 201

So, there are 201 terms in this series. Now we can find the sum using the arithmetic series sum formula:

Sum = (n / 2) * (first term + last term)

Sum = (201 / 2) * (4 + 1004)

Sum = (201 / 2) * 1008

Sum = 201 * 504

Sum = 101304

So, the sum of the series is 101304. Awesome job, everyone! We’ve successfully tackled all four math problems.

Conclusion

Wow, you guys did fantastic! We’ve worked through some challenging problems involving arithmetic series and sums. Remember, the key to solving these kinds of problems is to break them down into smaller, manageable steps. Whether it's finding the common difference, determining the number of terms, or applying the sum formula, each step gets us closer to the solution.

Keep practicing these techniques, and you'll become even more confident in your math skills. And remember, math can be fun when we approach it with curiosity and a willingness to learn. Keep up the great work, and I’ll see you in the next challenge!