Why do I keep getting aphids after I spray with insecticides?  

Plant sucking insects always have a ready food supply.  As long as the plant is there, the aphids will return.  Predatory insects, like ladybugs and praying mantis, must have a ready food supply of aphids or they will not stick around.  When you spray with insecticide, it kills both the aphids as well as any predatory insects that might be there trying to help you out.  Once the predatory insects are gone, but the plant food supply is still there, the aphids will return first.  The predators must have a very high population of aphids before they will return, but once the population of aphids is high enough, the predators will return.

Secondly, aphids born in spring are all females capable of reproducing more females without contact with males.  It is as if they were born pregnant.  They reproduce quickly and could produce billions of aphids per month, if they all lived.  Predatory insects are not that fast.  Ladybugs might only have one or two generations per year.  They must have the population of aphids first, then they have to reproduce.  You can see that the aphids can gain the upper hand quickly.  Our normal response is to spray again, which kills any new predators, thus starting the cycle over again.

There is a plant called butterfly weed.  (Read more about it at http://en.wikipedia.org/wiki/Butterfly_weed).  Not only does it attract butterflies, but it attracts aphids.  In fact it attracts aphids so strongly that many people plant it at a distance in their gardens simply to keep aphids away from their other plants.  As long as you leave this pest attractant around, your yard should be filled with plenty of predatory insects to help you keep the rest of the garden aphid free.