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Question

If in a progression $a subscript 1 end subscript comma a subscript 2 end subscript comma horizontal ellipsis comma$ etc., $left parenthesis a subscript r end subscript minus a subscript r plus 1 end subscript right parenthesis$ bears a constant ratio with $a subscript r end subscript cross times a subscript r plus 1 end subscript$ , then the terms of the progression are in

A.P.
G.P.
H.P.
None of these

The correct answer is: H.P.

$fraction numerator a subscript r end subscript minus a subscript r plus 1 end subscript over denominator a subscript r end subscript blank a subscript r plus 1 end subscript end fraction equals k$ (constant)
$rightwards double arrow fraction numerator 1 over denominator a subscript r plus 1 end subscript end fraction minus fraction numerator 1 over denominator a subscript r end subscript end fraction equals k$
$rightwards double arrow fraction numerator 1 over denominator a subscript 1 end subscript end fraction comma fraction numerator 1 over denominator a subscript 2 end subscript end fraction comma horizontal ellipsis comma fraction numerator 1 over denominator a subscript n end subscript end fraction$ are in A.P.
$rightwards double arrow a subscript 1 end subscript comma a subscript 2 end subscript comma a subscript 3 end subscript comma horizontal ellipsis comma$ are in H.P.

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