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Question Number 5723 Answers: 0 Comments: 2

(1/2)+(1/4)+(1/8)+....+(1/2^n )=1−(1/2^n ) P r o v e the above for integral n≥1.

$$\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{4}}+\frac{\mathrm{1}}{\mathrm{8}}+....+\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{n}} }=\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{n}} } \\ $$$$\mathrm{P}\:\:\:\mathrm{r}\:\:\:\mathrm{o}\:\:\:\mathrm{v}\:\:\:\mathrm{e}\:\mathrm{the}\:\mathrm{above}\:\mathrm{for}\:\mathrm{integral}\:\mathrm{n}\geqslant\mathrm{1}. \\ $$

Question Number 5722 Answers: 0 Comments: 2

Prove by mathematical induction that tbe following formula is correct for all positive integers n: ((2),(2) ) + ((3),(2) ) + ((4),(2) ) +...+ (((n+1)),(( 2)) ) = (((n+2)),(( 3)) )

$$\mathrm{Prove}\:\mathrm{by}\:\boldsymbol{\mathrm{mathematical}}\:\boldsymbol{\mathrm{induction}} \\ $$$$\mathrm{that}\:\mathrm{tbe}\:\mathrm{following}\:\mathrm{formula}\:\mathrm{is}\:\mathrm{correct} \\ $$$$\mathrm{for}\:\mathrm{all}\:\mathrm{positive}\:\mathrm{integers}\:\mathrm{n}: \\ $$$$\begin{pmatrix}{\mathrm{2}}\\{\mathrm{2}}\end{pmatrix}\:+\begin{pmatrix}{\mathrm{3}}\\{\mathrm{2}}\end{pmatrix}\:+\begin{pmatrix}{\mathrm{4}}\\{\mathrm{2}}\end{pmatrix}\:+...+\begin{pmatrix}{\mathrm{n}+\mathrm{1}}\\{\:\:\:\mathrm{2}}\end{pmatrix}\:=\begin{pmatrix}{\mathrm{n}+\mathrm{2}}\\{\:\:\:\mathrm{3}}\end{pmatrix} \\ $$

Question Number 5718 Answers: 0 Comments: 2

Two angles can be added and subtracted. Can we multiply them also? For example x radians × y radians = xy (radians)^2 ? What will be the meaning of (radians)^2 (square radians)?

$$\mathrm{Two}\:\mathrm{angles}\:\mathrm{can}\:\mathrm{be}\:\mathrm{added}\:\mathrm{and}\:\mathrm{subtracted}. \\ $$$$\mathrm{Can}\:\mathrm{we}\:\mathrm{multiply}\:\mathrm{them}\:\mathrm{also}? \\ $$$$\mathrm{For}\:\mathrm{example}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{x}\:\mathrm{radians}\:×\:\mathrm{y}\:\mathrm{radians}\:=\:\mathrm{xy}\:\left(\mathrm{radians}\right)^{\mathrm{2}} \:? \\ $$$$\mathrm{What}\:\mathrm{will}\:\mathrm{be}\:\mathrm{the}\:\mathrm{meaning}\:\mathrm{of}\:\:\left(\mathrm{radians}\right)^{\mathrm{2}} \\ $$$$\left(\mathrm{square}\:\mathrm{radians}\right)? \\ $$

Question Number 5715 Answers: 1 Comments: 0

Differentiate x^x from the first principle. please help

$${Differentiate}\:\:\:{x}^{{x}} \:\:\:{from}\:{the}\:{first}\:{principle}. \\ $$$$ \\ $$$${please}\:{help} \\ $$

Question Number 5712 Answers: 0 Comments: 3

Find the value of x 2^x = 4x workings is needed please.

$${Find}\:{the}\:{value}\:{of}\:{x}\: \\ $$$$ \\ $$$$\mathrm{2}^{{x}} \:=\:\mathrm{4}{x} \\ $$$$ \\ $$$${workings}\:{is}\:{needed}\:{please}. \\ $$

Question Number 5674 Answers: 0 Comments: 0

Find the value of x x^((x + 2)) = (x + 2)^x Please help.

$${Find}\:{the}\:{value}\:{of}\:{x}\: \\ $$$$ \\ $$$${x}^{\left({x}\:+\:\mathrm{2}\right)} \:=\:\left({x}\:+\:\mathrm{2}\right)^{{x}} \\ $$$$ \\ $$$${Please}\:{help}. \\ $$

Question Number 5673 Answers: 2 Comments: 1

Using inductive method. prove that .. 7^(2n) + 16n − 1 is divisible by 4 please help.

$${Using}\:{inductive}\:{method}.\:{prove}\:{that}\:.. \\ $$$$ \\ $$$$\mathrm{7}^{\mathrm{2}{n}} \:+\:\mathrm{16}{n}\:−\:\mathrm{1}\:{is}\:{divisible}\:{by}\:\mathrm{4} \\ $$$$ \\ $$$${please}\:{help}. \\ $$

Question Number 5672 Answers: 1 Comments: 0

Differentiate ((lnx)/e^x ) fom the first principle. Please help me.

$${Differentiate}\:\:\:\:\frac{{lnx}}{{e}^{{x}} }\:\:\:\:{fom}\:{the}\:{first}\:{principle}. \\ $$$$ \\ $$$${Please}\:{help}\:{me}. \\ $$

Question Number 5685 Answers: 1 Comments: 0

Find the value of x . 9^x = 6^x + 4^x Please help.

$${Find}\:{the}\:{value}\:{of}\:{x}\:. \\ $$$$ \\ $$$$\mathrm{9}^{{x}} \:=\:\mathrm{6}^{{x}} \:+\:\mathrm{4}^{{x}} \\ $$$$ \\ $$$${Please}\:{help}. \\ $$

Question Number 5684 Answers: 0 Comments: 1

Find the value of x . x^((x + 2)) = (x + 2)^x Thanks for your help.

$${Find}\:{the}\:{value}\:{of}\:{x}\:.\: \\ $$$$ \\ $$$${x}^{\left({x}\:+\:\mathrm{2}\right)} \:=\:\left({x}\:+\:\mathrm{2}\right)^{{x}} \\ $$$$ \\ $$$${Thanks}\:{for}\:{your}\:{help}. \\ $$

Question Number 5695 Answers: 1 Comments: 5

Question Number 5692 Answers: 0 Comments: 2

f(x)=e^x g(x)=ln x h(x)=x A line L is perpendicular to h(x) at point P(x,y) and extends and disects f(x) and g(x). The length of L between f(x) and g(x) is r. When is r minimum?

$${f}\left({x}\right)={e}^{{x}} \\ $$$${g}\left({x}\right)=\mathrm{ln}\:{x} \\ $$$${h}\left({x}\right)={x} \\ $$$$ \\ $$$$\mathrm{A}\:\mathrm{line}\:{L}\:\mathrm{is}\:\mathrm{perpendicular}\:\mathrm{to}\:{h}\left({x}\right)\:\mathrm{at}\:\mathrm{point} \\ $$$${P}\left({x},{y}\right)\:\mathrm{and}\:\mathrm{extends}\:\mathrm{and}\:\mathrm{disects}\:{f}\left({x}\right)\:\mathrm{and}\:{g}\left({x}\right). \\ $$$$\mathrm{The}\:\mathrm{length}\:\mathrm{of}\:{L}\:\mathrm{between}\:{f}\left({x}\right)\:{and}\:{g}\left({x}\right) \\ $$$$\mathrm{is}\:{r}.\:\mathrm{When}\:\mathrm{is}\:{r}\:\mathrm{minimum}? \\ $$

Question Number 5705 Answers: 1 Comments: 0