Animated Solution for Mathematics - Sequence and Series: If $x={\sum }_{n=0}^{\infty }{a}^n,y={\sum }_{n=0}^{\infty }{b}^n,z={\sum }_{n=0}^{\infty }{c}^n$, where $a,b,c$ are in A.P. and $|a|<1,|b|<1,|c|<1,abc\ne 0$, then

Imagine you are standing at the edge of an infinite staircase. Each step you take is smaller than the last, shrinking by a constant ratio. This is the essence of an infinite geometric series.

In our problem, we are given three such series: $x={\sum }_{n=0}^{\infty }{a}^n$, $y={\sum }_{n=0}^{\infty }{b}^n$, and $z={\sum }_{n=0}^{\infty }{c}^n$.

At first glance, these look like abstract symbols, but they are actually elegant, compact representations of infinite sums.

Let us focus on $x$. When we expand the summation, we see $x=1+a+a^2+a^3+\dots$.

This is the classic definition of a Geometric Progression (G.P.) where the first term is $1$ and the common ratio is $a$. The constraint $|a|<1$ is our green light; it tells us the series converges to a finite sum.

Using the formula $S=\frac{A}{1-R}$, we immediately find:

This is the key that unlocks the entire problem. By symmetry, we can write:

Now, we need to connect these to the condition that $a,b,$ and $c$ are in Arithmetic Progression (A.P.). The A.P. condition is $2b=a+c$.

However, our current equations are in terms of $x,y,$ and $z$. We need to flip the perspective. Rearranging $x=\frac{1}{1-a}$, we get $1-a=\frac{1}{x}$, which simplifies to:

Similarly, we find $b=1-\frac{1}{y}$ and $c=1-\frac{1}{z}$. We have successfully built a bridge between the world of G.P. sums and the world of A.P. sequences.

Now, let us perform the substitution into the A.P. condition $2b=a+c$. Substituting our expressions, we get:

Expanding the left side gives $2-\frac{2}{y}$, and combining the right side gives $2-\frac{1}{x}-\frac{1}{z}$. Look at the beauty of the algebra: the constant $2$ appears on both sides and cancels out perfectly!

We are left with $-\frac{2}{y}=-\frac{1}{x}-\frac{1}{z}$. Multiplying by $-1$, we arrive at the elegant result:

This is the definitive signature of an Arithmetic Progression. It tells us that the reciprocals $\frac{1}{x},\frac{1}{y},\frac{1}{z}$ are in A.P.