Intro to arithmetic progressions

so if you see a set of numbers like this and if someone asks you are these numbers in arithmetic progression the only thing you need to check is if you jump from one number to the next is the jump size remaining the same that's it that's the only condition you to care about and let's see over here here the jump size is plus one plus one plus one again plus one over here and plus one again over here and I'm gonna trust that this person who put the dot dot dot will continue to follow this rule and therefore I'll say oh yeah this is an arithmetic progression because the jump size is a constant here the numbers are just natural numbers 1 2 3 4 5 so the set of natural numbers are in arithmetic progression now let's let's go get a few more if I ask you let's say 1 3 5 just three numbers are they in arithmetic progression what will you do the only thing that defines an arithmetic progression is is the jump size constant so you will jump and you'll notice that okay that's a plus 2 jump here there's another plus 2 jump here so the jump size does not need to be 1 it can be 2 but it's still and it can only be 3 numbers it doesn't have to be a long sequence then these three numbers are an arithmetic progression okay what if I have something like this is this an arithmetic progression I know only one thing to check this plus 3 jumps is 3 again jumps is 3 again I'm gonna say this is an AP so maybe we should put it like that's an AP this is an AP this is an AP a B is a short-form we used to say that these numbers are in arithmetic progression let's do this 1 4 9 16 is this in AP let's see there is a +3 over here and there's a +5 over here I don't have to check any further this is not an AP no matter what happens over here this cannot be 1 because the jump size is different so the jump sir is need not be 1 or 2 it just has to be a constant but the moment it changes across then we know this is not an AP let's see what more we can have let's say that we have three five eight 11 ah is this an AP who here that jump size is two plus two over here it's plus three that's it I'm done I don't need to check any further I know that this is not an AP so not an AP what if I had eight Phi 2 minus one minus four so first you notice that this is reducing right so 8 - 5 - 2 that isn't matter my jump can be a negative jump even though I'm drawing it like this in my mind I'm actually going left in the number line right so I jump a minus 3 to get here I jump another minus 3 to get here another 2 and then minus 1 so minus 3 to get here and another minus 3 to get here so this is a arithmetic progression because the jump size is remaining a constant so you now know that the jump sir is need not be a positive number it can be a negative number for all we know let's look at one more seven seven seven seven is this even a sequence it's just the same number repeating is this an arithmetic progression I mean I'm just gonna do the same thing I'm not gonna think I have to jump a plus 0 or a minus 0 so plus zero and a plus zero in other words I'm standing on the same spot on the number line and I just standing over there and jumping in the same spot stationary jumping but it's is changing it's not so like we said the only thing we check is is the jump size changing its the term says changing its not then you say this is an arithmetic progression so in summary if you just look very quickly you can see that the number can start wherever it wants to right it can be 1 it can be for it even can be a negative number where you start it doesn't have to be a positive number so if I make up some some sequence like this - 3 - 0 - 3 - how much will that be 6 - 9 and so on this is an arithmetic sequence because you have your plus three to get here plus three again and so on so you can start wherever you want to positive negative zero this could have been the arithmetic sequence could have started right here these numbers will also be in arithmetic progression arithmetic sequence is just another word for numbers there are an arithmetic progression so a sequence is just a set of numbers right so this is what you have notice that also the common difference can be positive no that's a bad example to show let's say somewhere over here yeah it can be positive it can be zero and it can be negative so the only thing you need to know to solve any problem in arithmetic progressions is this there are set of numbers which are the same jump size between them can start wherever they want to can end wherever they want to are called or said to be an arithmetic progression anything else can be derived from this core idea