If X = sum_{n=0}^infty a^n,Y = sum_{n=0}^inft | vedclass

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(D) Given that $X = \sum_{n=0}^\infty a^n = \frac{1}{1-a}$,$Y = \sum_{n=0}^\infty b^n = \frac{1}{1-b}$,and $Z = \sum_{n=0}^\infty c^n = \frac{1}{1-c}$

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Harmonic progression

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(D) Given that $X = \sum_{n=0}^\infty a^n = \frac{1}{1-a}$,$Y = \sum_{n=0}^\infty b^n = \frac{1}{1-b}$,and $Z = \sum_{n=0}^\infty c^n = \frac{1}{1-c}$.Since $a, b, c$ are in arithmetic progression,we have $2b = a + c$.We need to check if $X, Y, Z$ are in harmonic progression,which requires $\frac{2}{Y} = \frac{1}{X} + \frac{1}{Z}$.Substituting the values: $\frac{1}{X} = 1-a$,$\frac{1}{Y} = 1-b$,and $\frac{1}{Z} = 1-c$.Then $\frac{1}{X} + \frac{1}{Z} = (1-a) + (1-c) = 2 - (a+c)$.Since $a+c = 2b$,this becomes $2 - 2b = 2(1-b) = 2(\frac{1}{Y}) = \frac{2}{Y}$.Thus,$X, Y, Z$ are in harmonic progression.

If $X = \sum_{n=0}^\infty a^n$,$Y = \sum_{n=0}^\infty b^n$,and $Z = \sum_{n=0}^\infty c^n$,where $a, b, c$ are in arithmetic progression and $|a| < 1, |b| < 1, |c| < 1$,then $X, Y, Z$ are in:

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