1. Key Concepts and Formulas

• Sum of an Infinite Geometric Progression (G.P.): For a geometric series with first term $A$ and common ratio $R$, if $|R| < 1$, the sum to infinity is given by $S_\infty = \frac{A}{1-R}$.
• Arithmetic Progression (A.P.): Three numbers $p, q, r$ are in A.P. if $2q = p + r$.
• Harmonic Progression (H.P.): Three non-zero numbers $p, q, r$ are in H.P. if their reciprocals $\frac{1}{p}, \frac{1}{q}, \frac{1}{r}$ are in A.P. This implies $\frac{2}{q} = \frac{1}{p} + \frac{1}{r}$.

2. Step-by-Step Solution

Step 1: Express $x, y, z$ using the sum of infinite G.P. formula. The given sums are infinite geometric series with first term $1$ (since $a^0=1, b^0=1, c^0=1$) and common ratios $a, b, c$ respectively. The condition $|a|<1, |b|<1, |c|<1$ ensures that these series converge. $x = \sum\limits_{n = 0}^\infty {{a^n}} = 1 + a + a^2 + \dots = \frac{1}{1-a} \quad (\text{Equation 1})$ $y = \sum\limits_{n = 0}^\infty {{b^n}} = 1 + b + b^2 + \dots = \frac{1}{1-b} \quad (\text{Equation 2})$ $z = \sum\limits_{n = 0}^\infty {{c^n}} = 1 + c + c^2 + \dots = \frac{1}{1-c} \quad (\text{Equation 3})$

Step 2: Express $a, b, c$ in terms of $x, y, z$. We rearrange the equations from Step 1 to isolate $a, b, c$. From Equation 1: $x = \frac{1}{1-a} \implies 1-a = \frac{1}{x} \implies a = 1 - \frac{1}{x} \quad (\text{Equation 4})$ From Equation 2: $y = \frac{1}{1-b} \implies 1-b = \frac{1}{y} \implies b = 1 - \frac{1}{y} \quad (\text{Equation 5})$ From Equation 3: $z = \frac{1}{1-c} \implies 1-c = \frac{1}{z} \implies c = 1 - \frac{1}{z} \quad (\text{Equation 6})$

Step 3: Use the A.P. condition for $a, b, c$. The problem states that $a, b, c$ are in A.P. This means the middle term $b$ is the arithmetic mean of $a$ and $c$. $2b = a + c$ Substitute the expressions for $a, b, c$ from Equations 4, 5, and 6 into this condition. $2\left(1 - \frac{1}{y}\right) = \left(1 - \frac{1}{x}\right) + \left(1 - \frac{1}{z}\right)$

Step 4: Simplify the equation and identify the progression of $x, y, z$. Let's simplify the equation obtained in Step 3: $2 - \frac{2}{y} = 1 - \frac{1}{x} + 1 - \frac{1}{z}$ $2 - \frac{2}{y} = 2 - \frac{1}{x} - \frac{1}{z}$ Subtract 2 from both sides: $-\frac{2}{y} = -\frac{1}{x} - \frac{1}{z}$ Multiply by -1: $\frac{2}{y} = \frac{1}{x} + \frac{1}{z}$ This equation shows that the reciprocals of $x, y, z$ are in A.P. ($\frac{1}{x}, \frac{1}{y}, \frac{1}{z}$ are in A.P. because $\frac{2}{y} = \frac{1}{x} + \frac{1}{z}$). By the definition of a Harmonic Progression, if the reciprocals of a sequence of numbers are in A.P., then the numbers themselves are in H.P.

3. Common Mistakes & Tips

• Forgetting the Convergence Condition: The condition $|a|, |b|, |c| < 1$ is vital for the sum of infinite G.P. to exist. Always ensure this condition is met or stated.
• Confusing A.P. and H.P. Definitions: Remember that H.P. is defined by the A.P. relationship of the reciprocals. A common mistake is to apply the A.P. condition directly to $x, y, z$.
• Algebraic Errors: Careless algebraic manipulation, especially with fractions, can lead to incorrect conclusions. Double-check rearrangements.

4. Summary

The problem involves converting sums of infinite geometric progressions into expressions for $x, y, z$ in terms of $a, b, c$. By rearranging these expressions to find $a, b, c$ in terms of $x, y, z$, and then applying the given condition that $a, b, c$ are in A.P., we derived a relationship between $x, y, z$. This derived relationship, $\frac{2}{y} = \frac{1}{x} + \frac{1}{z}$, is the defining property of a Harmonic Progression, meaning $x, y, z$ are in H.P.

The final answer is $\boxed{H.P.}$, which corresponds to option (D).