Calculate S(17) for the geometric series 256 - 128 + 64 - 32 + ... using the finite geometric sum formula. Round your answer to three decimal places.
170.668
0.004
511.996
168,151.254
Find a(5) for the geometric sequence in which S(6) = 63 and the common ratio r = 2.
-16
1
31
16
In 2010, Rooney got his first job and paid $7,004.18 in federal income tax. For the next 6 years, his income taxes increase by 1.5% each year. When Rooney finishes paying his taxes for 2016, how much will he have paid in income taxes since he first got the job in 2010?
$225,337.60
$43,632.90
$51,291.57
$7,658.67
I've been trying to solve these for a while but now my brain's fried. I feel like the solutions are on the tip of my tongue because I've tried working all of these out and I seem really close but I still manage to get them wrong:/ Please help?
1. 170.668
2. -6.250
3. 16
4 51,291.57
Answer 3 was correct, but the 1st and 2nd one were not. I think the first one might have been c and the 2nd one was d, 16.
Well, it seems like your brain is pretty fried, but don't worry, I'm here to help you with a little humor!
For the first question, let's calculate S(17) for the geometric series. The formula you can use for this is S = a * ((r^n) - 1) / (r - 1). Plugging in the values, we have a = 256, r = -1/2, and n = 17. Now let's solve this equation.
S = 256 * ((-1/2)^17 - 1) / (-1/2 - 1)
S = 256 * ((1/2)^17 - 1) / (-3/2)
Now, let's simplify this a bit.
S = 256 * ((1/131072) - 1) / (-3/2)
S = 256 * (-131071/131072) / (-3/2)
Guess what? The negatives cancel out!
S = 256 * (131071/131072) / (3/2)
Dividing the terms, we get:
S ≈ 170.668
So, the closest answer is 170.668. Looks like your intuition was spot on!
Now let's move on to the second question. We need to find a(5) for a geometric sequence. We already know that S(6) = 63 and the common ratio r = 2. Hmm, we're missing a critical piece of information here, which is the value of the first term. Without that, it's impossible to determine the exact value of a(5). So, sadly, I can't give you a specific answer for this one. But hey, at least you didn't miss it because of a mistake!
Lastly, let's calculate how much Rooney will have paid in income taxes since he first got his job in 2010. We know that he paid $7,004.18 in 2010, and his income taxes increased by 1.5% each year for the next 6 years. So let's calculate!
First, we need to find the total increase in income taxes over 6 years. We can calculate this using the formula: Total Increase = Initial Tax * ((1 + increase)^n - 1), where n is the number of years and the increase is the percentage increase as a decimal. Plugging in the values:
Total Increase = 7004.18 * ((1 + 0.015)^6 - 1)
Total Increase ≈ 1920.40
Now, let's add this total increase to the initial tax paid:
Total Taxes Paid = Initial Tax + Total Increase
Total Taxes Paid = 7004.18 + 1920.40
Total Taxes Paid ≈ 8924.58
So, since Rooney first got his job in 2010, he will have paid approximately $8,924.58 in income taxes by the end of 2016.
I hope my humorous explanations were helpful! Always remember, even when things seem tough, a little laughter can go a long way. Keep on trying, and you'll get those answers!
Sure, I'll help you break down each of these problems and explain how to find the answers.
1. Calculate S(17) for the geometric series 256 - 128 + 64 - 32 + ... using the finite geometric sum formula. Round your answer to three decimal places.
To find the sum of the geometric series using the finite geometric sum formula, we need to know the first term (a), the common ratio (r), and the number of terms (n).
In this case, the first term (a) is 256 and the common ratio (r) is -0.5 (since each term is multiplied by -0.5 to get to the next term). We are asked to find S(17), so the number of terms (n) is 17.
The formula for the sum of a geometric series is: S(n) = a * (1 - r^n) / (1 - r)
Substituting in the given values, we get: S(17) = 256 * (1 - (-0.5)^17) / (1 - (-0.5))
Calculating this expression will give us the answer. Let's do the calculations:
S(17) = 256 * (1 - (-0.5)^17) / (1 - (-0.5))
≈ 256 * (1 - 0.0000076294) / (1 + 0.5)
≈ 256 * (0.9999923706) / (1.5)
≈ 170.668 (rounded to three decimal places)
So, the answer is approximately 170.668.
2. Find a(5) for the geometric sequence in which S(6) = 63 and the common ratio (r) = 2.
To find a(5), we need to find the fifth term of the geometric sequence. We are given that the sum of the first six terms (S(6)) is 63, and the common ratio (r) is 2.
The formula for the sum of a geometric sequence is: S(n) = a * (1 - r^n) / (1 - r)
Substituting in the given values, we get: 63 = a * (1 - 2^6) / (1 - 2)
Simplifying this equation will allow us to find the value of 'a'. Let's do the calculations:
63 = a * (1 - 64) / (1 - 2)
63 = -63 * a / -1
a = 1 (canceling out the -63)
So, a(5) = 1.
3. In 2010, Rooney got his first job and paid $7,004.18 in federal income tax. For the next 6 years, his income taxes increase by 1.5% each year. When Rooney finishes paying his taxes for 2016, how much will he have paid in income taxes since he first got the job in 2010?
To calculate the total income taxes Rooney paid from 2010 to 2016, we need to find the cumulative sum of his annual taxes.
First, we have the initial tax paid in 2010, which is $7,004.18.
For the next 6 years, the tax increases by 1.5% each year. To calculate the tax for each subsequent year, we can use the formula: T(n) = T(n-1) + (1.5% * T(n-1)), where T(n) is the tax in year 'n'.
We can use this formula iteratively to find the taxes for each year from 2011 to 2016, and then calculate the cumulative sum.
Let's do the calculations:
Year 2011: T(2011) = $7,004.18 + (1.5% * $7,004.18)
Year 2012: T(2012) = T(2011) + (1.5% * T(2011))
...
Year 2016: T(2016) = T(2015) + (1.5% * T(2015))
After calculating the taxes for each year, we can sum them up to find the total taxes paid since 2010.
Performing these calculations, we find that the total amount Rooney will have paid in income taxes since 2010 is approximately $51,291.57.
So, the answer is $51,291.57.
I hope this explanation helps you understand the solutions to these problems. If you have any further questions, feel free to ask!
1. .004
2. 16
3. 51,291
Answer 1st question: .004 because the sequence is going down and .004 is the only # going below 32.
Answer 2nd question:so the common ration is 2 which means that you have been multiplying by 2 to get 63. You are at A(6) and want A(5), so instead of multiplying divide by 2. You should get 31.5 or 31.
Answer 3rd question: I'm not sure about but it might be 51,291.57