[Music] so ladies and gentlemen are you ready ladies and gentlemen may i have your attention please it's time for the final countdown [Music] the show starts in [Music] go all right guys good day everyone again this is sergio venezuela junior from sdo qc new era high school so welcome to another math 21 with sir g another great and lesson to be discussed without further ado join me so before we start guys let's discuss uh the learning goals of today's lesson so after this lesson you're going to differentiate you can differentiate the arithmetic sequence to arithmetic series for number two you can define and illustrate an arithmetic series for number three you can identify each element of the formula of course the arithmetic series and for number four you can solve problem involving arithmetic series or some of the series again welcome to arithmetic series again this is surgery so let's discuss the formula so in arithmetic series guys we have two formulas to use no so let me name this the first formula and this is the second formula just formula okay so [Music] so since we are in arithmetic series guys so when we talk about arithmetic series we are looking for the sum of the entire series no some of the series so every time uh you've heard or you've read the word series we are talking about the sun so that's why in my formula in our formula brother we use s for the arithmetic series since we are looking for the sum of the entire series no so you know that is the number of terms all right all right so all n belongs to number of terms of course the uh numerical two here so that is a constant no so constant sugar so always divide by two then again young torendito beside the first term or a sub one is a constant so constant i just then plus that is again constant all right then a quantity of n minus 1 again 1 here is a constant then we multiply to the common difference so that is our first formula no so again we read this formula as the sum of the entire series is always equal to the half of the number of terms multiply to twice of the first term plus a quantity of the number of terms minus one times the common difference formula then multiply to two times a sub one plus a quantity of n minus one times d so parang i know essence not [Music] [Music] as the sum of the series then is equal to again your n is the number of terms so half of the number of terms then multiply to twice of the first term since a1 a sub 1 is the first term then plus a quantity of the number of terms which is n minus 1 then times your common difference okay and that is the first formula for the arithmetic series so quantize the second formula so theta guys again the same note so we use s as the sum of the entire series again n here is the number of terms for n and if you could notice guys made you make little formulas term and the last term so therefore i first term and last term so indeed first term and last term uh well that is about to use the first formula okay so again so we're going to read the second formula as the sum of the entire series is equal to half of the number of terms multiplied to the sum between the first and the last term okay so guys all right so let's try to solve problems [Music] okay so we have to read and comprehend all right so we are asked here to find the sum so you know problem nothing guys entire series for milan it denotes s sub n so therefore young sub and nothing is so let's continue reading so find the sum of the first 10 terms so now we gain a number of terms how many terms do we have we have 10 terms therefore our n is equal to 10 so guys so let's continue so find the sum of the first ten terms of the arithmetic series all right 4 that is the first term then we have 10 that is the second term 16 that is the third term etc until we arrive at the tenth term so we are looking for the uh sum of the ten terms we are not looking for the tenth term so guys is arithmetic sequence all right so that's uh basically the difference between arithmetic series and arithmetic sequence no society sequence guys focused is until the last term so that's that's basically the difference between the two okay all right so let's continue so let's solve this problem no so candida young question meeting first formula at second formula okay so tandan young second formula nathan guys uh it requires us to have both the first term and the last term attends a problem so therefore we have no choice but but to use the first pattern all right so again based on the problem guys entire series then of course of course your first element owns a sequence i4 and 16 and so on no so final move on a common difference so let's recall starting arithmetic sequence so in a common difference guys by subtracting from the right side right side towards to its left until we arrive at the first term no so 16 minus 10 that would be 6 all right then ten minus four that would be six so since a common difference something that goes up and nothing guys we expect that the difference uh are common or no so that's why our d here is six now here are the given then we're going to use the first formula guys again we're not inviting a meeting in second formula because the last term is missing no my first serve long time okay so let's state the formula guys get into time and solve lag in a problem you have to get the given then state the formula then bhagawatayama well guided n is read as the sum of the series so this time we are looking for the sum of the ten terms is equal to uh the number of terms over two or half of the number of terms then multiply to twice of the first term then plus a quantity of the number of terms minus one times the common difference so that's how to read and i want you to so again the sum of the series is equal to the half of the number of terms then multiply two twice of the first term then plus a quantity of the number of terms minus one times the common difference so i guess and then nothing we have to solve and to do that the first thing that we're going to do guys is to substitute or plug in our given so we have n so all n must be replaced by 10 so if you could notice all n here was uh was replaced by 10 basically okay then the next thing that we're going to do is to replace our a sub 1 which is our first term so on the given the first term is equal to 4 that's why we have 4 here sir that is the constant two in the formula okay so don't be um then of course 10-1 again your n is 10 of course then minus 1 is constant then your common difference is 6. all right so after we plug in all the givens we arrive at s sub 10 or the sum of the 10 terms is equal to half of 10 then multiply to twice of 4 then plus 10 minus 1 times 6. so let's simplify so what is 10 over 2 so that is equal to five then two times four that is equal to eight then ten minus one that is equal to nine then we just bring down six okay then let's simplify so what is nine times six so nine times six is 54 then bringing down eight so eight plus 54 that is 62 all right then 62 times five we will arrive at 310 so therefore the sum of the first ten terms of the series whose first term is four second term is ten and third term is sixteen is three hundred and ten let's proceed on our second problem okay uh how many terms no how many terms so therefore your number of terms all right so how many terms again reading can't stand alone reading with comprehension is very very good okay so how many terms again that is n n is missing okay how many terms of the series 20 so this is your first term 18 that is your second term and 16 your third term so how many terms must be added you know must be added so that the sum oh yeah guys i'll be gaining some the sum will be negative 100 so therefore the sum of these uh series is equal to negative 100. in total is negative so 2018-16 and so okay all right so so your first element not in 20 and second element not in 18 the third element not in 16 and so on so again um or sequence arithmetic sequence young given elements is in decreasing format automatic the common difference is negative now take note huh arithmetic sequence and we are under arithmetic series given elements is in decreasing format so automatic uncommon difference not a negative on the other hand in increasing format the common difference is always positive okay so take note of that okay so let's subtract again not to subtract so 16 minus 8 10 and 16 minus 18 so 16 minus 18 guys nothing absolute value than 16 that is 16 and an absolute value now minus 18 that is positive 18 absolute value so therefore absolute value now minus eighteen so therefore you and your kings are good so sixteen minus eighteen so that is equal to negative two all right so instead of positive two or series no arithmetic sequence no or arithmetic series uh automatic uncommon difference not in a negative so 18 minus 16 that is 2 so negative 2. twenty minus eighteen that is two so negative two okay so i hope it negative two hung up in common difference okay i am so let me clear this uh so in explaining so just uh for recall on how to get the common difference and i know that indiana last term but to use the first formula okay so the sum of the series is equal to half of the number of terms twice the first term plus a quantity of the number of terms minus one times the common difference then let's plug in all the given so from the given benefits entire series with its which is negative 100 all right and since young and nothing guys are unknown so n over 2 then twice of the first term so your first term nothing guys 20 so i plug in 20 then plus again n minus 1 n is missing kaya and parensha then plug in athenian common differences which is negative two okay so i hope okay again in and so in this case young sometimes here is a negative 100 so 2 times negative 100 guys that is equal to negative 200 right and come on then what is left after so that is equal to 40 then again this is multiplication of monomial versus binomials therefore we distribute no so negative two multiply two positive n so that is negative two n okay so take note of the sign again when we multiply unlike sine always gives us negative result while like sign gives us positive results that's why negative 2 times negative 1 is positive 2 then after doing so we arrive at negative 200 equals to n then multiply to 40 minus 2n plus 2. now relax very good again pause the video rewind and re-watch until you master the skill then what's next so we will bring down negative 200. and then 40 plus 2 that is equal to 42 then ethereum minus 2n okay i hope all right so again pause the video rewind rewatch okay again this is monomial versus binomial so multiply only taiyo distribute nothing on n okay so n times negative 2 n so unlike sine the answer is negative 2 n squared so take note we add exponents in multiplication then n multiply to 42 that is 42n then if you could notice we arrive at negative 200 equals negative 2 n squared plus 42 n at i'm sure highest degree now polynomial during grade nine and so i'm sure you don't then let's rearrange the equation so let's equate the entire equation by zero so i will transfer all elements on the right side to the left and take note when we do that we always change the sign so from negative to n square we arrived at positive 2n square from positive 42 we arrive at negative 42 n and of course we simply copy negative 200 and equals to zero and let's examine the equation so market in you guys not in [Music] so 2 divided by 2 that is equal to 1 then 1 times n squared so we will arrive at n squared the negative 42 divided by 2 so that is negative 21 then just bring down n of course negative 200 divided by 2 that is negative 100 all right and of course zero by two that is equal to zero guys okay [Music] okay so because this is a real problem so we have to answer the problem in real as well okay so let's factor this out so let's factor n squared minus 21 n minus 100 equals to zero okay so every time that the last term is negative guys okay kappa last term not n i negative so automatic guys yes every time and i say every time that is always so every time they own last term nothing negative automatic [Music] yes positive this is a factor in 100 i 25 times 4 so 25 times or actually marami and 50 times 2 1 times 100 two times 50 etc until we arrive at 25 times four kasinga angus young difference between the factors is 21 at it spinal condition because 25 minus 4 is 21. so therefore ito now questions good question that's why we arrive at n plus 4 times n minus 25 equals to zero then zero property equating n plus four equals to 0 and n minus 5 n minus 25 equals to 0. then let's simplify you know so we can use property of equality or we can simply transfer positive 4 to the right side then we then let's continue so we will arrive at n minus 25 equals to zero the same thing no let's transfer negative 25 to the right side and we will arrive at positive 25 which is our final answer so therefore so let's conclude there are 25 terms in the given series okay so 25 terms para angmaging sum i negative 100 say in young double checking okay so i hope nintendo had nothing guys again hindi you may always pause the video rewind re-watch use your pen and paper okay so let's try another problem that is no problem nothing guys okay and so let's find the sum of the odd integers [Music] [Music] 246 18 hungar 100 and of course i want to 100 so we expect the meron thai on 50 odd numbers so therefore the integers in pinakusa [Music] one three five seven etc until you arrive at 99. if you're going to get the cardinality of o you will get 50 and okay all right and again guys our first term here is in one okay the first term is one okay so let's try to solve this problem so therefore we can use the second formula uh second formula can i use the first formula okay let's see let's solve this problem okay so let's analyze again the problem now based on accounting numbers no one to 100 there are 50 even and 50 odd number yeah so young first term not in canada one attempt last term 1999 last term so therefore we can use the second formula and so sobrang dalilang or sum num 50 terms is equal to you know half num 50 then of course let's plug in the value of a sub 1 or first term so your first term is equal to 1 and your last term is equal to 99 then get the sum that is equal to 100 and the half of 50 that is equal to 25 then 25 times 100 and we will arrive at 2500 or 2500 therefore the sum of the 50 terms of the series whose elements are add numbers from 1 to 99 inclusive no is equal to 2 500. formula foreign [Music] [Music] i won uncommon difference a2 uncommon difference a2 1 3 5 7 99 increasing so therefore uncommon difference not an a positive they subtract seven minus five two five minus three two three minus one two okay uncommon difference not in a two okay and [Music] then 50 divided by 2 that is equal to 25 then 2 times 1 that is equal to 2 then 50 minus 1 that is equal to 49 then just bring down 2. then 49 times 2 that is 98 plus 2 so young samyang i 100 then 25 times 100 again we will arrive at 2500 to keep the sum of the 50 odd integers from 1 to 99 inclusive all right hi and so again you may always pause the video rewind re-watch then again use your pen and paper somebody this is sir hill thank you keep safe guys