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kirill [66]
3 years ago
11

What is the sum of the first 70 consecutive odd numbers? Explain.

Mathematics
1 answer:
expeople1 [14]3 years ago
5 0

The sum of the first n odd numbers is n squared! So, the short answer is that the sum of the first 70 odd numbers is 70 squared, i.e. 4900.

Allow me to prove the result: odd numbers come in the form 2n-1, because 2n is always even, and the number immediately before an even number is always odd.

So, if we sum the first N odd numbers, we have

\displaystyle \sum_{i=1}^N 2i-1 = 2\sum_{i=1}^N i - \sum_{i=1}^N 1

The first sum is the sum of all integers from 1 to N, which is N(N+1)/2. We want twice this sum, so we have

\displaystyle 2\sum_{i=1}^N i = 2\cdot\dfrac{N(N+1)}{2}=N(N+1)

The second sum is simply the sum of N ones:

\underbrace{1+1+1\ldots+1}_{N\text{ times}}=N

So, the final result is

\displaystyle \sum_{i=1}^N 2i-1 = 2\sum_{i=1}^N i - \sum_{i=1}^N 1 = N(N+1)-N = N^2+N-N = N^2

which ends the proof.

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From 19 measurements of a pressure controller, the mean operating pressure is 4.97 kPa with a standard deviation of 0.0461 kPa.
Semenov [28]

Answer:

4.97-2.10\frac{0.0461}{\sqrt{19}}=4.95    

4.97+2.10\frac{0.0461}{\sqrt{19}}=4.99    

So on this case the 95% confidence interval would be given by (4.95;4.99)  

Step-by-step explanation:

Previous concepts

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".

The margin of error is the range of values below and above the sample statistic in a confidence interval.

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

\bar X =4.97 represent the sample mean

\mu population mean (variable of interest)

s=0.0461 represent the sample standard deviation

n=19 represent the sample size  

Confidence interval

The confidence interval for the mean is given by the following formula:

\bar X \pm t_{\alpha/2}\frac{s}{\sqrt{n}}   (1)

In order to calculate the critical value t_{\alpha/2} we need to find first the degrees of freedom, given by:

df=n-1=19-1=18

Since the Confidence is 0.95 or 95%, the value of \alpha=0.05 and \alpha/2 =0.025, and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-T.INV(0.025,18)".And we see that t_{\alpha/2}=2.10

Now we have everything in order to replace into formula (1):

4.97-2.10\frac{0.0461}{\sqrt{19}}=4.95    

4.97+2.10\frac{0.0461}{\sqrt{19}}=4.99    

So on this case the 95% confidence interval would be given by (4.95;4.99)    

5 0
3 years ago
Wendell wants to make a wooden lid for a decorative box. The lid needs to be 612inches wide and 812inches long with sides 2 inch
andriy [413]

Answer:

Step-by-step explanation:

A

3 0
2 years ago
Find a positive angle less than one revolution around the unit circle that is co-terminal with the given angle: 52pi/5
Ludmilka [50]
We know that
Two angles are said to be co-terminal <span>if they have the same initial side and </span>
<span>the same terminal side. 
</span>
(52π/5)-----> 10.4π<span>
so
</span>(52π/5)-5*2π------> (2π/5)

the answer is
the positive angle less than one revolution around the unit circle that is co-terminal with angle of 52π/5 is 2π/5

7 0
3 years ago
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Step-by-step explanation:

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3 years ago
Solve the following quadratic by<br> completing the square.<br> y = x2 - 6x + 7
kobusy [5.1K]

Answer:

x = 5  or   x = 1

Step-by-step explanation:

 x² - 6x + 7 = 0

x² - 6x = -7

x² - 6x + 9 = -7 + 9      1/2 of the x term than square it and add it the both sides.

(x - 3)(x - 3) = 2

( x -3)² = 2

\sqrt{(x - 3)^2 = \sqrt{2}

(x -  3) = ± 2

x - 3 = 2  or  x - 3 = -2

x = 5     or  x = 1

7 0
3 years ago
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