Gear 5 and 6 got slower in bike? Yesterday I pedaled reverse (one circle) while getting on my bike, I heard a sound at the chains and after that the bike speed on my 5th and 6th gear got slower. Before I used to pedal less and the bike would go faster but now I have to pedal alot faster for the bike to move ahead. What is the problem? Should I visit the cycle repair shop for fixing or can I fix it by myself?
Sounds to me like your derailleur is not properly shifting (to the smaller cog) in those gears you mentioned. 

First thing to check (imo):
Put the bike in a bike stand or position it upside down, shift onto smallest rear cog. 
Watch the cassette/chain whilst pedaling with one hand and shifting with the other hand. Check if the chain properly shifts each time you press the shift button with your hand. Do some shifts require more than 1 'click' of the shifter before the chain moves? Are any cogs being skipped? Are the outer most cogs (biggest/smallest) on your cassette ever reached by the chain (if you cannot shift into these (biggest/smallest cog) your derailleur limits will be improperly adjusted or (for biggest cog) cable tension may be too low.

Probably a cable issue (a cable in bad condition (even though it might look like nothing is wrong with it) can cause issues similar to the one you are describing). Especially when downshifting (to a smaller cog) this can cause issues since when up-shifting you 'pull' the derailleur to the next cog using the cable (you can determine the force on the cable with your thumb so even with a little more cable friction you'll likely be able to up-shift just fine) and when downshifting you are relying on the derailleur spring to move the derailleur into the desired position. 

The issue could be with your shifter also but it's less likely in my experience.

A few things to check:


Remove cable from rear derailleur by undoing the cable clamp bolt, then pull the cable taught with one hand whilst shifting with the other hand. See if the cable moves smoothly through all the gears (both in up-shifting and in down-shifting). If the cable is not moving smoothly in either direction, remove the cable housing from the shift cable and check again, if the problem (cable not moving smoothly) is gone without the cable housing you know the issue lies with said cable housing.
Check the cable housing for any exterior damage (cuts, sharp bends etc.). If you have a lot of spare length (cable housing) and if the damage is near the ends of the housing you could consider shortening the housing to remove the damaged length. If that is not the case replace the entire piece of housing which shows the exterior damage. If the cuts are not deep enough to cut through the plastic/rubber of the cable housing then the cuts might not be a problem. As soon as the cuts result in exposed metal they will rust (if they are not stainless steel) which could very likely cause (undesired) extra cable friction.
Remove the shift cable from the housing and check it for any fraying/rust/loose wires sticking out etc. If the cable is rusty you could clean it with for example steel wool and lubricate it but your best option is to replace it (preferably with a stainless steel variant). If the cable is frayed or damaged in another way definitely replace it.
Check the condition of the cable housing end caps, are they bent/deformed in any way? If so definitely replace them.
Check (without shifter cable attached) the rear derailleur, check how smooth it moves (by pushing it from side to side with your hand), lubricate the linkage pins.
Whilst the shift cable is removed from the rear derailleur, move the derailleur from one side to the other side of the cassette by hand (whilst turning the pedals with your other hand), check if the chain can reach all the cogs (if the limits are set incorrectly the chain might A: not move far enough to reach the biggest/smallest cog or might not move onto said cogs far enough causing it to 'fall off' (jump to the cog next to it) when putting any pedaling force on the chain or B:move too far causing the chain to fall off the cassette (in between frame/cassette or in between hub/spokes and cassette). You should be able to move the derailleur by hand (when the chain is on the biggest/smallest gear) such that the center of the upper jockey wheel of the derailleur is approximately 1-2mm further than the biggest/smallest cog itself. (please check the Park tool video -linked below- on derailleur adjustment for more info on this)


Cable end caps:



I would personally suggest replacing the following:


inner shifter cable
shifter cable housing
shifter cable housing end caps


The cable housing and end caps shouldn't cost much, especially when bought in bulk (10+ meter). You can also buy sets which are meant to replace the cables on just one bike, but they are much more expensive and I have not personally noticed a difference in quality. 

There are special pliers for cutting cable housing which are great but you can work with normal side cutters also, just have to make sure the cable is cut straight and when cutting brake housing cables make sure the 'spiral' (metal spiral that runs through the core of the cable housing) doesn't have any sharp pieces sticking out after cutting (you might have to cut a few times to get it perfect).

You can definitely do it yourself. Derailleur will have to be readjusted (cable tension) after you've replaced the cable, there are a lot of tutorials online on how to do that.

Some tutorials you might find useful:
Replace shift cables:


                
            

Adjust derailleur:

Note: when replacing the cable you'll only have to re-adjust the cable tension after having installed the new cable (assuming all (other) prior adjustments were correct), it can't hurt to check the other adjustments whilst your at it (how to do that is shown in the video :) )

https://www.parktool.com/blog/repair-help/rear-derailleur-adjustment

Adjusting derailleurs is very doable as long as your cables are in good condition, if they are not then you might be adjusting for an hour and you won't be able to get it to shift correctly (similar to your situation now).

Alternatively:
Your derailleur or derailleur hanger could be bent. You can check this with a derailleur alignment gauge (or in very severe cases it can be seen with the naked eye). On steel frames you can usually bend the derailleur hanger back if it's bent. On aluminum frames you'll have to replace the derailleur hanger (separate piece) if it's bent. This seems quite unlikely based on your description of the cause (backpedaling) but I thought I'd mention it just in case..

Here is some info on derailleur alignment:

Derailleur hanger alignment: https://bit.ly/2TVoYUJ

DIY derailleur hanger alignment tool: https://bit.ly/2RRvbyz

Align derailleur hanger without special tools: https://bit.ly/2TTF6Gj

How to straighten a bent derailleur: https://bit.ly/3aL3VdQ

Some more useful info:

https://www.parktool.com/blog/repair-help/front-derailleur-adjustment
(thank you for the suggestion @Argenti Apparatus )

Hope this helps, let me know if you have any questions.