Arrange the APs in the increasing order of their common differences (top to bottom).

4, -1, -6, -11..

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4, 1, -2, -5..

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5, 4, 3, 2 ..

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7, 9, 11, 13..

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7, 11, 15, 19..

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Solution

The common difference of an AP is the difference between any two consecutive terms of the AP.

In the AP 4, -1, -6, -11,...
$a_{1} = 4$ and $a_{2} = - 1.$
But, $d = a_{2} - a_{1} = - 1 - 4 = - 5.$

In the AP 5, 4, 3, 2, ...
$a_{1} = 5$ and $a_{2} = 4.$
But, $d = a_{2} - a_{1} = 4 - 5 = - 1.$

In the AP 7, 9, 11, 13, ...
$a_{1} = 7$ and $a_{2} = 9.$
But, $d = a_{2} - a_{1} = 9 - 7 = 2.$

In the AP 7, 11, 15, 19, ...
$a_{1} = 7$ and $a_{2} = 11.$
But, $d = a_{2} - a_{1} = 11 - 7 = 4.$

In the AP 4, 1, -2, -5, ...
$a_{1} = 4$ and $a_{2} = 1.$
But, $d = a_{2} - a_{1} = 1 - 4 = - 3.$

Since $- 5 < - 3 < - 1 < 2 < 4,$ we have the required order.

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