hi guys it's me turgon in our today's video we will talk about the arithmetic series automatic series is considered as the partial sum of a given number or a given number of terms in a given arithmetic sequence so right now we have here two different formulas that we will be using in this video wherein we have here S Sub n this S Sub n stands for the sum of a given arithmetic series or some of the terms in a given analytic sequence so for the first Formula S Sub n is equal to n times a sub 1 plus a sub n over 2. where inner n is the number of terms a sub 1 is the first term a sub n is considered as the last term in a given series for the next Formula we have your a sub S Sub n is equal to n over 2. times 2 a sub 1 plus n minus 1 times d so the difference between these two formula or concrete in the meeting is that you can use the first Formula if given the universe and last term of the given series here in a man uh you can use this formula if I'm given Lang I a sub 1 and n and your common difference so let's solve a problem we have here find the sum of the first 20 terms so we have here first 20 terms [Music] first 20 terms of a given series were in term our first term is five well a sub 20 is 62. now in this case guys your a sub 1 is equal to five this a sub 20 will be considered as the last term in a given series meaning your a sub 20 is also equal to a sub m and that is equal to 62 meaning if we will choose among these two formula much better to use S Sub n okay let's write down the formula again in this paper we have S Sub n is equal to n times a sub 1 plus a sub n over two so what's next is that we will use this formula this s of n will become S Sub 20. because all we need to do is to get the sum of the first 20 terms so this is 20. and then year n here is equal to 20. meaning for this variable n it will become 20 times your a sub 1 which is 5. plus your a sub n or a sub 20 or a sub 20 is equal to 62 over 2. okay simplify this it will become 20 times 5 plus 6 is 2 that is equal to 67 over 2. so as you can see guys when you multiply this to numerator numerator tapos your denominator is here is one one times two it will become 20 times six over seven a six seven over two we're in instead of multiplying 20 by 67 much better if we will simplify first twenty and two so we can cancel out two cancel at point it will become ten so what we have now is simply 10 plus 6 times 10 times 67 meaning your S Sub 20 or the sum of the first 20 terms of the given sequence are series or in the first term in the 20th term 62. their sum is equal to 670 because we have 10 times 67 and this is the answer for the first problem now let's continue for the second problem guys here's the second problem what we have here is we need to find the sum of the first 40 terms of the arithmetic Series 2 5 8 and 11 and so on now as you can see we only have here a sub 1 we don't have the last term of the given sequence meaning among the two different formulas that we have kanina we will be using this formula okay so I will try to rewrite the formula we have S Sub n is equal to n over 2 times 2 a sub 1 plus n minus 1 times d okay so let's list down all the needed variables here your a sub 1 is definitely two now for the variable n as you can see we have here first 40 terms meaning your n is equal to 40. for the common difference d so you can easily identify this one because this one is an easy type of arithmetic sequence 5 minus two is three eight minus five is three eleven minus eight is three therefore their common difference is three now after getting these variables we are now ready to use this formula your S Sub n will become S Sub 20 Over N over 2 ah sorry this is 40 guys S Sub 40 my fault guys this is Asap 40 is equal to 40 over 2 times 2 times 2 times your a sub 1 is 2 this is 2 plus we have here n minus one here n is 40 then you have minus 1 here times d which is equal to 3. simplify first we have S Sub 4 d 40 divided by 2 is 20. now let me use the parentheses 2 times 2 is 4. Plus 40 minus 1 is 39 so we have 39 times 3. okay so what we have now first is to simplify this you're 39 times 3 so we have 3 times 10 which is 27 so this is 7 then carry two three times three is nine plus two which is eleven meaning this is 117 so we have S of 40. is equal to 20 times 4 plus 117 right and this one we have 20 times 4 plus 17 which is equal to one hundred twenty one so what we need to do now is to multiply this so your S Sub 40 is equal to 121 times 20. bring down zero two times one is two two times two is four one times one is two meaning the sum of the first 40 terms of the given series is two thousand four hundred twenty okay guys so I hope guys to learn something from this video on how to do the sum of the given arithmetic series using these two formulas so I hope you like this video so if you're new to my channel don't forget to like And subscribe but hit the Bell button for you to be updated latest uploads again it's me teacher gone bye