For what value of $x$ are $x+2,4x-1$ and $6x+3$ three consecutive terms of an AP?

In an AP the middle term is the average of its neighbours, so $2(4x-1)=(x+2)+(6x+3)$. That gives $8x-2=7x+5$, so $x=7$ and the terms are 9, 27, 45 with $d=18$. Forgetting to double the middle term gives $x=-2$, and slips in collecting the constants give 3 or 5.