<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] 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 $\frac12(2^{50}-0^{50})=2^{49}$.
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