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2 years ago

Find the sum of the following without actually adding: 1+3+5+7+9+11+13+15

Mathematics

2 answers:

-Dominant- [34]2 years ago

7 0

Answer:

Step-by-step explanation:

The following numbers are successive odd numbers.

The sum of successive odd numbers (starting from 1) is always the square of number of terms you are adding.

Hear there are 8 terms. Square of 8 is 64.

Hence the sum of this numbers is<u> 64</u>.

I am Lyosha [343]2 years ago

6 0

use this formula Sn=n/2[2a+(n-1)d]

here,d=a2-a1 and n= no. Of terms

Step-by-step explanation:

S8=8/2[2(1)+(8-1)2]

S8=64

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xP(x)00.2510.0520.1530.55Find the standard deviation of this probability distribution. Give your answer to at least 2 decimal pl

Nookie1986 [14]

The Solution:

Given:

Required:

Find the standard deviation of the probability distribution.

Step 1:

Find the expected value of the probability distribution.

$E(x)=\mu=\sum_{i\mathop{=}0}^3x_iP_(x_i)$ $\begin{gathered} \mu=(0\times0.25)+(1\times0.05)+(2\times0.15)+(3\times0.55) \\ \\ \mu=0+0.05+0.30+1.65=2.0 \end{gathered}$

Step 2:

Find the standard deviation.

$Standard\text{ Deviation}=\sqrt{\sum_{i\mathop{=}0}^3(x_i-\mu)^2P_(x_i)}$ $=(0-2)^2(0.25)+(1-2)^2(0.05)+(2-2)^2(0.15)+(3-2)^2(0.55)$ $=4(0.25)+1(0.05)+0(0.15)+1(0.55)$ $=1+0.05+0+0.55=1.60$

Thus, the standard deviation is 1.60

Answer:

1.60

8 0

1 year ago

1 - Graph a line with a slope of 4

AVprozaik [17]

Y=4x+2 that my equation now graph it

4 0

3 years ago

=) The perimeter of a rectangular field is 336 m.

goldenfox [79]

Answer:

Length of the field: 94 m

Width of the painting: 61 cm

Step-by-step explanation:

Use the perimeter formula, P = 2l + 2w, to find the length:

Plug in the perimeter and width into the equation:

P = 2l + 2w

336 = 2l + 2(74)

336 = 2l + 148

188 = 2l

94 = l

So, the length of the field is 94 m.

To find the width of the painting, use the area formula, A = lw

Plug in the area and length into the equation:

A = lw

5795 = 95w

61 = w

So, the width of the painting is 61 cm.

Length of the field: 94 m

Width of the painting: 61 cm

4 0

2 years ago

Find the Fourier series of f on the given interval. f(x) = 1, ?7 < x < 0 1 + x, 0 ? x < 7

Zolol [24]

$f(x)=\begin{cases}1&\text{for }-7$

The Fourier series expansion of $f(x)$ is given by

$\dfrac{a_0}2+\displaystyle\sum_{n\ge1}a_n\cos\frac{n\pi x}7+\sum_{n\ge1}b_n\sin\frac{n\pi x}7$

where we have

$a_0=\displaystyle\frac17\int_{-7}^7f(x)\,\mathrm dx$
$a_0=\displaystyle\frac17\left(\int_{-7}^0\mathrm dx+\int_0^7(1+x)\,\mathrm dx\right)$
$a_0=\dfrac{7+\frac{63}2}7=\dfrac{11}2$

The coefficients of the cosine series are

$a_n=\displaystyle\frac17\int_{-7}^7f(x)\cos\dfrac{n\pi x}7\,\mathrm dx$
$a_n=\displaystyle\frac17\left(\int_{-7}^0\cos\frac{n\pi x}7\,\mathrm dx+\int_0^7(1+x)\cos\frac{n\pi x}7\,\mathrm dx\right)$
$a_n=\dfrac{9\sin n\pi}{n\pi}+\dfrac{7\cos n\pi-7}{n^2\pi^2}$
$a_n=\dfrac{7(-1)^n-7}{n^2\pi^2}$

When $n$ is even, the numerator vanishes, so we consider odd $n$ , i.e. $n=2k-1$ for $k\in\mathbb N$ , leaving us with

$a_n=a_{2k-1}=\dfrac{7(-1)-7}{(2k-1)^2\pi^2}=-\dfrac{14}{(2k-1)^2\pi^2}$

Meanwhile, the coefficients of the sine series are given by

$b_n=\displaystyle\frac17\int_{-7}^7f(x)\sin\dfrac{n\pi x}7\,\mathrm dx$
$b_n=\displaystyle\frac17\left(\int_{-7}^0\sin\dfrac{n\pi x}7\,\mathrm dx+\int_0^7(1+x)\sin\dfrac{n\pi x}7\,\mathrm dx\right)$
$b_n=-\dfrac{7\cos n\pi}{n\pi}+\dfrac{7\sin n\pi}{n^2\pi^2}$
$b_n=\dfrac{7(-1)^{n+1}}{n\pi}$

So the Fourier series expansion for $f(x)$ is

$f(x)\sim\dfrac{11}4-\dfrac{14}{\pi^2}\displaystyle\sum_{n\ge1}\frac1{(2n-1)^2}\cos\frac{(2n-1)\pi x}7+\frac7\pi\sum_{n\ge1}\frac{(-1)^{n+1}}n\sin\frac{n\pi x}7$

3 0

2 years ago

Three times a number, decreased by 8, is the same as twice the number, increased by 15. Find

natima [27]

3x - 8 = 2x + 15
Add 8 to both sides
3x = 2x + 23
Subtract 2x from both sides
x = 23

3 0

3 years ago