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3 years ago

14

Which of the following sequences are not geometric? (check all that apply) a. 2,10,50,250,1250 b. 1,4,9,16,25,36 c. -4,-2,-1,-0.

5,-0.25,-0.125 d. 1,1,2,3,5,8,13,21

1 answer:

Romashka [77]3 years ago

8 0

A is geometric because each number is multiplied by 5.

B is not geometric because it is an arithmetic sequence.

C is a geometric sequence because each number is divided by 2.

D is neither geometric not arithmetic because there is no common ratio and there is not a pattern being added or subtracted to each number.

So, your answer should be B and D.

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2 years ago

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Harman [31]

Answer:

See below for proof.

Step-by-step explanation:

<u>Given</u>:

$y=\left(x+\sqrt{1+x^2}\right)^m$

<u>First derivative</u>

$\boxed{\begin{minipage}{5.4 cm}\underline{Chain Rule for Differentiation}\\\\If $f(g(x))$ then:\\\\$\dfrac{\text{d}y}{\text{d}x}=f'(g(x))\:g'(x)$\\\end{minipage}}$

<u />

<u /> $\boxed{\begin{minipage}{5 cm}\underline{Differentiating $x^n$}\\\\If $y=x^n$, then $\dfrac{\text{d}y}{\text{d}x}=xn^{n-1}$\\\end{minipage}}$

<u />

$\begin{aligned} y_1=\dfrac{\text{d}y}{\text{d}x} & =m\left(x+\sqrt{1+x^2}\right)^{m-1} \cdot \left(1+\dfrac{2x}{2\sqrt{1+x^2}} \right)\\\\ & =m\left(x+\sqrt{1+x^2}\right)^{m-1} \cdot \left(1+\dfrac{x}{\sqrt{1+x^2}} \right) \\\\ & =m\left(x+\sqrt{1+x^2}\right)^{m-1} \cdot \left(\dfrac{x+\sqrt{1+x^2}}{\sqrt{1+x^2}} \right)\\\\ & = \dfrac{m}{\sqrt{1+x^2}} \cdot \left(x+\sqrt{1+x^2}\right)^{m-1} \cdot \left(x+\sqrt{1+x^2}\right)\\\\ & = \dfrac{m}{\sqrt{1+x^2}}\left(x+\sqrt{1+x^2}\right)^m\end{aligned}$

<u>Second derivative</u>

<u />

$\boxed{\begin{minipage}{5.5 cm}\underline{Product Rule for Differentiation}\\\\If $y=uv$ then:\\\\$\dfrac{\text{d}y}{\text{d}x}=u\dfrac{\text{d}v}{\text{d}x}+v\dfrac{\text{d}u}{\text{d}x}$\\\end{minipage}}$

$\textsf{Let }u=\dfrac{m}{\sqrt{1+x^2}}$

$\implies \dfrac{\text{d}u}{\text{d}x}=-\dfrac{mx}{\left(1+x^2\right)^\frac{3}{2}}$

$\textsf{Let }v=\left(x+\sqrt{1+x^2}\right)^m$

$\implies \dfrac{\text{d}v}{\text{d}x}=\dfrac{m}{\sqrt{1+x^2}} \cdot \left(x+\sqrt{1+x^2}\right)^m$

$\begin{aligned}y_2=\dfrac{\text{d}^2y}{\text{d}x^2}&=\dfrac{m}{\sqrt{1+x^2}}\cdot\dfrac{m}{\sqrt{1+x^2}}\cdot\left(x+\sqrt{1+x^2}\right)^m+\left(x+\sqrt{1+x^2}\right)^m\cdot-\dfrac{mx}{\left(1+x^2\right)^\frac{3}{2}}\\\\&=\dfrac{m^2}{1+x^2}\cdot\left(x+\sqrt{1+x^2}\right)^m+\left(x+\sqrt{1+x^2}\right)^m\cdot-\dfrac{mx}{\left(1+x^2\right)\sqrt{1+x^2}}\\\\ &=\left(x+\sqrt{1+x^2}\right)^m\left(\dfrac{m^2}{1+x^2}-\dfrac{mx}{\left(1+x^2\right)\sqrt{1+x^2}}\right)\\\\\end{aligned}$

$= \dfrac{\left(x+\sqrt{1+x^2}\right)^m}{1+x^2}\right)\left(m^2-\dfrac{mx}{\sqrt{1+x^2}}\right)$

<u>Proof</u>

$(x^2+1)y_2+xy_1-m^2y$

$= (x^2+1) \dfrac{\left(x+\sqrt{1+x^2}\right)^m}{1+x^2}\left(m^2-\dfrac{mx}{\sqrt{1+x^2}}\right)+\dfrac{mx}{\sqrt{1+x^2}}\left(x+\sqrt{1+x^2}\right)^m-m^2\left(x+\sqrt{1+x^2\right)^m$

$= \left(x+\sqrt{1+x^2}\right)^m\left(m^2-\dfrac{mx}{\sqrt{1+x^2}}\right)+\dfrac{mx}{\sqrt{1+x^2}}\left(x+\sqrt{1+x^2}\right)^m-m^2\left(x+\sqrt{1+x^2\right)^m$

$= \left(x+\sqrt{1+x^2}\right)^m\left[m^2-\dfrac{mx}{\sqrt{1+x^2}}+\dfrac{mx}{\sqrt{1+x^2}}-m^2\right]$

$= \left(x+\sqrt{1+x^2}\right)^m\left[0]$

$= 0$

8 0

1 year ago

In January. The temperature in Fargo, North Dakota is -2 F. On the same day, the temperature in Cheyenne, Wyoming was 1 F. Write

zaharov [31]

Answer:

(a) $1 > -2$ or $-2 < 1$

(b) $2 > 1$ or $1 < 2$

Step-by-step explanation:

Given

$N = -2F$ --- North Dakota

$C = 1F$ --- Cheyenne, Wyoming

Solving (a): Inequality to compare both temperatures

From the given temperatures, we can conclude that:

$C > N$ or $N < C$

Because

$1 > -2$ i.e. 1 is greater than -2

or

$-2 < 1$ i.e. -2 is less than 1

Solving (b): Inequality to compare the absolute values of the temperatures

We have:

$N = -2F$

$C = 1F$

The absolute values are:

$|N| = |-2|$

$|N| =2$

$|C| = |1|$

$|C| = 1$

By comparison:

$|N| > |C|$ or $|C| < |N|$

Because

$2 > 1$ i.e. 2 is greater than 1

or

$1 < 2$ i.e. 1 is less than 2

4 0

2 years ago