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Question 1: if the arithmetic mean of two numbers is twice of their geometric mean, their ratio of sum of numbers to the difference of numbers equals?

Question 2: if the quadratic equation:

$(b^2+c^2)x2-2(a+b)cx+(c^2+a^2)=0$

has equal roots then?what is its AP & GP?

Question 3: If the expansion of:

$(1+x)^{50}$

let S be the sum of the coefficient of the odd power of x, then S will be?

Please help with this problems in brief. -Thanks.

  • 0
    AP is Arithmetic Progression & GP is Geometric Progression.2012-10-03

3 Answers 3

0

if the quadratic equation:

$(b^2+c^2)x^2-2(a+b)cx+(c^2+a^2)=0$

Now we can write it as $\displaystyle (bx-c)^2+(cx-a)^2 = 0$

Means $bx-c = 0\Rightarrow bx = c$ and $cx-a = 0 \Rightarrow cx=a$

So $\displaystyle x = \frac{c}{b} = \frac{a}{c}\Rightarrow c^2 = ab$

So $a,b,c$ are in $\bf{Geometric\; Progression.}$

3

Question 1:

Arithmetic mean: $\frac{a+b}{2}$

Geometric mean: $\sqrt{ab}$

So the condition becomes $\frac{a+b}{2}=2\sqrt{ab}$. Square both sides to get $ \frac{a^2+2ab+b^2}{4}=4ab $ which, assuming $b\ne0$, results in $ \left(\frac ab\right)^2-14\frac ab+1=0 $ and $ \frac ab=7\pm4\sqrt{3} $ Then we can compute $ \frac{a+b}{a-b}=\frac{\frac ab+1}{\frac ab-1}=\frac{4\pm2\sqrt{3}}{3\pm2\sqrt{3}}\frac{3\mp2\sqrt{3}}{3\mp2\sqrt{3}}=\pm\frac{2\sqrt{3}}{3} $

Question 3:

The sum of all the coefficients is $(1+1)^{50}=2^{50}$

The sum of the even coefficients minus the sum of the odd coefficients is $(1-1)^{50}=0^{50}$

The sum of the odd coefficients is $\frac12(2^{50}-0^{50})=2^{49}$.

1

$1.$ We can even do it without the quadratic formula. We have $a+b=4\sqrt{ab}$ and therefore
$(a+b)^2=16ab.$ Also, $(a-b)^2=(a+b)^2-4ab=(a+b)^2-\frac{1}{4}(a+b)^2=\frac{3}{4}(a+b)^2.$ Thus $\frac{(a+b)^2}{(a-b)^2}=\frac{4}{3},$ and therefore $\frac{a+b}{a-b}=\pm\frac{2}{\sqrt{3}}.$

$2.$ The roots are equal precisely if the discriminant is $0$, that is, if $4(a+b)^2c^2-4(b^2+c^2)(c^2+a^2)=0.$ Divide by $4$, expand everything, do the obvious cancellations. We get $2abc^2=c^4+a^2b^2,$ which can be rewritten as $(c^2-ab)^2=0.$ We conclude that $ab=c^2$. We cannot have $c=0$ and $b=0$, else we would not have a quadratic equation. We conclude that the sequence $b,c,a$ is a three-term geometric sequence. If $a\ne 0$, then $a,c,b$ is also a three-term geometric sequence.

$3.$ I recommend that you look at the solution by robjohn.

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    robjohn & An$d$re Nicolas --- Thanks for the help.2012-10-03