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Ahat [919]
3 years ago
11

. Evaluate the series 1 + 0.1 + 0.01 + . . .

Mathematics
2 answers:
EastWind [94]3 years ago
6 0

We can employ a simple repeated decimal trick:

x=1.111\ldots

0.1x=0.111\ldots

\implies x-0.1x=1\implies0.9x=1\implies x=\dfrac1{0.9}=\dfrac{10}9

###

Alternatively, we can compute the partial sum of the series.

\displaystyle S_n=\sum_{k=0}^n\dfrac1{10^k}

S_n=1+0.1+0.01+\cdots+\dfrac1{10^n}

0.1S_n=0.1+0.01+0.001+\cdots+\dfrac1{10^{n+1}}

\implies S_n-0.1S_n=0.9S_n=1-\dfrac1{10^{n+1}}

\implies S_n=\dfrac{10}9-\dfrac9{10^n}

As n\to\infty, the second term vanishes and we're left with \dfrac{10}9. Notice that this is really just a more formal version of the earlier trick.

amm18123 years ago
5 0

Answer:

1.11

Step-by-step explanation:

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vodomira [7]

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6 0
3 years ago
What are the coordinates of the x-intercepts of the parabola y = x² - 8x + 15?
tamaranim1 [39]

Answer:

(3, 0) and (5, 0)

Step-by-step explanation:

we have

y=x^{2}-8x+15

we know that

The x-intercepts are the values of x when the value of y is equal to zero

so

For y=0

x^{2}-8x+15=0

The formula to solve a quadratic equation of the form

ax^{2} +bx+c=0

is equal to

x=\frac{-b\pm\sqrt{b^{2}-4ac}} {2a}

in this problem we have

x^{2}-8x+15=0  

so

a=1\\b=-8\\c=15

substitute in the formula

x=\frac{-(-8)\pm\sqrt{-8^{2}-4(1)(15)}} {2(1)}

x=\frac{8\pm\sqrt{4}} {2}

x=\frac{8\pm2} {2}

x=\frac{8+2} {2}=5

x=\frac{8-2} {2}=3

so

x=3, x=5

therefore

The x-intercepts are (3,0) and (5,0)

4 0
3 years ago
Fred borrows $1120 at 19% intenest
myrzilka [38]

Answer:

$638.4

Step-by-step explanation:

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5 0
3 years ago
Which equation represents the function shown on the graph?<br> y=1/4x<br> y=1/2x<br> y=2x<br> y=4x
Alexus [3.1K]
Y=4x. y=mx+b m=slope b=y intercept m=4 because it goes up 4 over 1 and 4/1=4 b=0 because it started at 0, making the equation y=4x+0 or just y=4x
5 0
3 years ago
Read 2 more answers
Assume that the paired data came from a population that is normally distributed. using a 0.05 significance level and dequalsxmin
Artemon [7]
"<span>Assume that the paired data came from a population that is normally distributed. Using a 0.05 significance level and d = (x - y), find \bar{d}, s_{d}, the t-test statistic, and the critical values to test the claim that \mu_{d} = 0"

You did not attach the data, therefore I can give you the general explanation on how to find the values required and an example of a random paired data.

For the example, please refer to the attached picture.

A) Find </span><span>\bar{d}
You are asked to find the mean difference between the two variables, which is given by the formula:
\bar{d} =  \frac{\sum (x - y)}{n}

These are the steps to follow:
1) compute for each pair the difference d = (x - y)
2) sum all the differences
3) divide the sum by the number of pairs (n)

In our example: 
</span><span>\bar{d} =  \frac{6}{8} = 0.75</span>

B) Find <span>s_{d}
</span><span>You are asked to find the standard deviation, which is given by the formula:
</span>s_{d} =  \sqrt{ \frac{\sum(d - \bar{d}) }{n-1} }

These are the steps to follow:
1) Subtract the mean difference from each pair's difference 
2) square the differences found
3) sum the squares
4) divide by the degree of freedom DF = n - 1

In our example:
s_{d} = \sqrt{ \frac{101.5}{8-1} }
= √14.5
= 3.81

C) Find the t-test statistic.
You are asked to calculate the t-value for your statistics, which is given by the formula:
t =  \frac{(\bar{x} - \bar{y}) - \mu_{d} }{SE}

where SE = standard error is given by the formula:
SE =  \frac{ s_{d} }{ \sqrt{n} }

These are the steps to follow:
1) calculate the standard error (divide the standard deviation by the number of pairs)
2) calculate the mean value of x (sum all the values of x and then divide by the number of pairs)
3) calculate the mean value of y (sum all the values of y and then divide by the number of pairs)
4) subtract the mean y value from the mean x value
5) from this difference, subtract  \mu_{d}
6) divide by the standard error

In our example:
SE = 3.81 / √8
      = 1.346

The problem gives us <span>\mu_{d} = 0, therefore:
t = [(9.75 - 9) - 0] / 1.346</span>
  = 0.56

D) Find t_{\alpha / 2}
You are asked to find what is the t-value for a 0.05 significance level.

In order to do so, you need to look at a t-table distribution for DF = 7 and A = 0.05 (see second picture attached).

We find <span>t_{\alpha / 2} = 1.895</span>

Since our t-value is less than <span>t_{\alpha / 2}</span> we can reject our null hypothesis!!

7 0
3 years ago
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