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For what value of $k$ are $2k-7$, $k+5$ and $3k+2$ consecutive terms of an Arithmetic Progression?

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    Another useful thing to remember is that the sum of three consecutive terms of an arithmetic progression is equal to three times the middle term. If the terms can be taken out of order (i.e. they are three consecutive terms of a progression, but not in order) there is a second value of $k$ to be found.2012-10-30

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Hint: The numbers $a,b,c$ are consecutive terms of an AP if and only if $c-b=b-a$.