Sequences and Series - Relationship Between A.M. and G.M.

Definition of Arithmetic Mean (A.M.): For any two positive real numbers $a$ and $b$, the Arithmetic Mean is the value $A = \frac{a+b}{2}$. Visually, on a number line, $A$ represents the exact midpoint between the points $a$ and $b$.

Definition of Geometric Mean (G.M.): For any two positive real numbers $a$ and $b$, the Geometric Mean is the value $G = \sqrt{ab}$. Geometrically, if you have a rectangle with sides $a$ and $b$, $G$ is the side of a square that has the same area as that rectangle.

The A.M.-G.M. Inequality: For any two positive real numbers $a$ and $b$, the Arithmetic Mean is always greater than or equal to the Geometric Mean ($A \ge G$). On a coordinate plane, the value of $A$ will always be located at or to the right of $G$ on the horizontal axis.

Condition for Equality: The relationship $A = G$ occurs if and only if the two numbers are equal, i.e., $a = b$. If $a \neq b$, then $A$ is strictly greater than $G$ ($A > G$).

Geometric Visualization via Semicircle: Consider a semicircle with a diameter of length $a+b$. The radius of this circle is the A.M., which is $\frac{a+b}{2}$. If a perpendicular line is drawn from the point where segments $a$ and $b$ meet on the diameter to the circumference, the length of this perpendicular is the G.M., $\sqrt{ab}$. Since the radius is the maximum possible height within the semicircle, it visually demonstrates that A.M. $\ge$ G.M.

Quadratic Equation Connection: If $A$ and $G$ are the A.M. and G.M. of two numbers $a$ and $b$, then these numbers are the roots of the quadratic equation $x^2 - 2Ax + G^2 = 0$. This relates the means to the algebraic structure of polynomial roots.

Nature of Roots: Since $A \ge G$, the discriminant of the equation $x^2 - 2Ax + G^2 = 0$, which is $D = 4(A^2 - G^2)$, is always non-negative. This ensures that the numbers $a$ and $b$ are always real when $A$ and $G$ are given such that $A \ge G$.