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If $l=1+a+a^2+ \dots$, $m=1+b+b^2+ \dots$, and $n=1+c+c^2+ \dots$, where $\mid a \mid <1, \: \mid b \mid < 1, \: \mid c \mid <1$ and $a,b,c$ are in arithmetic progression, then $l, m, n$ are in

  1. arithmetic progression
  2. geometric progression
  3. harmonic progression
  4. none of these
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2 Answers

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1 votes
Harmonic Progression is the answer.

Let us take a = 0.1, b = 0.2, c = 0.3. Then l = 10/9, m = 10/8, n = 10/7 which are in harmonic progression.
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The formula for the sum of infinite geometric sequence is given by

$$1 + x + x^2 + x^3 + \cdots \cdots = \frac{1}{1-x}$$

 where  $\mid x \mid <1$

Now 

$$l=\frac{1}{1-a}$$

$$m=\frac{1}{1-b}$$

$$n=\frac{1}{1-c}$$

since $a$,$b$,$c$ are in AP let initial term is $s$ and common difference is $d$ then $a=s$, $b=s+d$, $c=s+2d$

$\frac{1}{l} =1-a=1-s $

$\frac{1}{m} =1-b=1-s-d $

$\frac{1}{n} =1-c=1-s-2d $

clearly $\frac{1}{l},\frac{1}{m},\frac{1}{n}$ are in A.P, with the common difference $-d$

$\implies l,m,n $ are in H.P.

The answer is OPTION C 

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