This fault is caused by a bent pin in the cartridge slot, its quite easy to repair and can be done by yourself, you do not need to pay some amateur to fix it, all it takes is a little patience.
Take a close look inside slot 1 and you will most likely see the problem. Below is what you need to repair it.
Nintendo DS Screwdriver.
2 small flat head screwdrivers.
Open the console carefully, the metal sheath covers the cartridge slot so you will have to remove this first.
Use 2 small screwdrivers to bend the pin back into shape, one is needed for leverage.
Put console back together. Takes about 20 mins.
Incidentaly, there are very few problems with this console, no power or power related problems such as not charging are caused by one or both of the surface mounted fuses ( f1/f2 ) going open circuit, these are quite near the metal sheath.
You should replace these if possible or you can bridge them, replacement is recommended as bridging may damage your console if there is a power surge so only use that solution as a temporary fix.