# If #a,b,c# are in arithmetic progression; #p,q,r# are in harmonic progression; #ap,bq,cr# are in geometric progression then prove that #a:b:c# is equal to #1/r:1/q:1/p#?

Please see below.

Multiplying (1) and (2), we get

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To prove that ( \frac{a}{b}:\frac{b}{q}:\frac{c}{r} = \frac{1}{r}:\frac{1}{q}:\frac{1}{p} ), we can start by expressing the terms of each progression:

Arithmetic progression: [ a = x ] [ b = x + d ] [ c = x + 2d ]

Harmonic progression: [ \frac{1}{p} = \frac{1}{x} ] [ \frac{1}{q} = \frac{1}{x + d} ] [ \frac{1}{r} = \frac{1}{x + 2d} ]

Geometric progression: [ ap = x \cdot \frac{1}{p} = 1 ] [ bq = (x + d) \cdot \frac{1}{q} = \frac{x + d}{x + d} = 1 ] [ cr = (x + 2d) \cdot \frac{1}{r} = \frac{x + 2d}{x + 2d} = 1 ]

So, ( ap = bq = cr = 1 ), which means ( ap, bq, cr ) are all equal to 1.

Therefore, ( \frac{a}{b}:\frac{b}{q}:\frac{c}{r} = 1:1:1 ), which is equivalent to ( 1/r:1/q:1/p ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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