For what value of $k$ are the number $x, 2 x + k$ and $3 x + 6$ are three consecutive term of an AP?

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Solution

For a set of numbers in AP, the common difference should be equal
$2 x + k - x = 3 x + 6 - (2 x + k)$
$x + k = 3 x + 6 - 2 x - k$
$x + k = x + 6 - k$
$x + k - x + k = 6$
$2 k = 6$
$k = 3$
Value of k should be $3$ .

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