<s>[INST] Arithmetic Mean & Geometric Mean 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.
 [/INST] $1.$ 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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