Chain skipping on some gears
As you asked for an explanation, why this skipping is happening, here it is:

For the beginning let's consider an unused chain on an unused sprocket. In this case the distance between the teeth of the sprocket is the same as the length of the chain links. Therefore, if you apply a force to the sprocket via the chain, the load is equally distributed over all teeth that are in contact with the chain. Due to the force the chain lengthens over time (not much, about 1-2% is enough to make the chain unuseable) which means that the chain links no longer fit the sprocket's teeth distance and also the load distribution isn't equal any more.

Instead, the tooth that is closest to the direction where the force comes from (the last one on the sprocket where the chain leaves towards the chain-blade and the first one one the chain-blade where the chain comes from the sprockets) gets the most load and therefore is wearing out a bit more than the other ones. As everything goes round and round and every tooth is in the position to wear out more than the others, the teeth will slightly adapt the valleys between them to fit the length of the chain.

If you replace now the chain by a new one while keeping the cassette the same (it would be the same if you replaced the cassette keeping the chain), then the chain doesn't fit any more to the cassette that has adapted to the length of the used chain links. If the length difference between old and new chain links is too much (as already said, some 1 or 2% are enough) the chain will slide up the shoulders of the teeth and skip into the next (or even the next-nearest) valley as soon as the applied force gets too high. One can make the components last somewhat longer by changing the chain early enough, but as both, chain and cassette are wearable parts they have to be replaced sooner or later.

It depends on different factors how fast this wear-out process will be:


Dirt and lubrication: Dust, rust, sand etc. act like grinding media which increase the friction between the chain and the sprocket and therefore accelerate the wear-out. A well cleaned and lubed drive train therefore lasts longer.
Load per tooth: The higher the force working on the sprocket and the lower the number of teeth over which this force is distributed, the faster the teeth are wearing out. Therefore it is usually the smallest sprockets on your cassette that at first show the skipping as they get quite a high load and have not much teeth which have to take this load.
Usage frequency of the sprocket or chain-blade: of course, a sprocket that is used more often wears out faster. This is the reason why for most people the middle sized chain-blade at their cranks will show skipping (if they ever come that far within the lifetime of their bike ;-).


This means that a road bike, where you normally don't have much dust and try to pedal rather smoothly without any rapid changes in force, can go several thousands of kilometers on one chain. With mountainbikes where you have the exact opposite (lots of dirt, often sharp load peaks when you go to rough terrain) the rule of thumb that I learned some years ago is to replace the chain every 2000km. Even then you will need a new cassette every second or third chain.