Sum to n terms of an arithmetic progression (2/2)

a factory-made 3000 toys in the first tier it made ninety-nine thousand toys in total increasing the production by 445 toys every year for how many years did the factory make the toys so what's happening is in the first year of its operation a company makes three thousand toys and every year after this it increases its production by 425 toys meaning that in the second year company would make four hundred and twenty-five toys more than it made in the first year so in the second year the total number of toys produced by the company would be three thousand four hundred and twenty five similarly in the third year of its operation it again increases the total number of toys by four hundred and twenty five as compared to the previous year so the total number of toys now produced by the company at the end of the third year would be three thousand eight hundred sorry 3850 toys and this process keeps on repeating until the total number of toys made by the company since the beginning of its operation comes out to be 99 thousand toys in total so what we have to figure out is for how many years did the company make these toys such that the some of the toys made every year amounts to ninety nine thousand so the number of toys made by this factory every year is an arithmetic progression since the difference between the number of toys in any two consecutive years is always going to be a constant so let's assume that this Factory operates for Enya's and it makes some amount of toys in its anythi now the sum of all the toys made by this factory up to its end atheer of operation comes out to be ninety nine thousand and we need to find out what this n is going to be so we already know that sum of n terms in an arithmetic aggression can be given by the average of the first and the eighth term times the number of terms in that progression so let's use this equation to figure out how many years it takes the company to make 99 thousand toys so we already know this sum of first n terms of the sequence we are also given the first term of the sequence as three thousand what we don't know is what should be the nth term of the sequence and how many terms does this sequence have so we have two unknowns right over here so how can we go about finding both of them so one way to think about it is if we could somehow figure out one of the unknowns in terms of the other one we would be able to find out this thing so let's use this pattern and see if we could relate these two unknowns so in the sequence every other term is D more than its previous term in other words if I have to get to my second term I'll add this D once to my first term if I have to get to my third term I'll add D twice to my first term and if I have to get to my init term I'll have to add D and minus 1 times to my first term which is a 1 so any init term in an AP is always the first term plus n minus 1 times the common difference so let's replace the value of a n in our equation over here and on doing that I'll get SN as n by 2 times and by 2 times 2 times a 1 plus n minus 1 times our common difference D so I think we have everything that we need now so why don't you pause this video and find out n all by yourself all right so let's do this together the sum of first n terms is given as 99,000 n is something that we need to figure out so I'll write that as this and then in the brackets we have 2 times a 1 which is 2 times 3,000 plus n minus 1 we still need to figure out n times the common difference which is 425 in this case so let's simplify this expression to figure out n I'll just make some space right over here okay so on multiplying both sides of this equation by two on the left hand side we'll have one lag 98,000 and on the right hand side we'll have n times two times three thousand s six thousand and if I distribute 425 over N and negative one I'll have negative four hundred and twenty five plus 425 times N or 425 n now we can simplify this further and add six thousand and negative 425 which comes out to be five thousand five hundred and seventy-five oh boy seems like we are getting ourselves a really big quadratic equation I wish I had simplified this in the previous step but anyway let's move on and let's simplify this further so on the right hand side we will now have five thousand five hundred and seventy-five n plus plus 425 N squared well this is really big so I'm going to get my calculator okay so I think 25 would be a factor of all these three terms so let's divide one lag 98,000 by 25 and we get seven thousand nine hundred and twenty was that 7920 or something else let me get that back okay so seven thousand nine hundred and twenty and on dividing 5575 by 25 will get two hundred and twenty three so here I can write 223 N and on dividing 425 by 25 I'll get 17 so this is 17 N squared okay so this is the simplest version because 17 is a prime number so on subtracting 7920 from both the sides we have 17 N squared plus 223 n minus seven thousand nine hundred and twenty equals to 0 so factorization seems a little tricky here I'm simply going to go ahead and use the quadratic formula according to which the roots of any quadratic equation can be given by minus b plus minus under root b square minus 4ac times 2a all you need to do is identify the correct coefficients here so here 17 is a 223 is B and negative 7 9 to 0 is our C so on substituting these values of ABC in this equation over here the roots of this quadratic equation comes out to be 16 and negative 495 wise 17 if you want to understand how did we get these values you can go ahead and watch some videos on quadratic equation and see how did we use quadratic formula to solve this so I'm going to go ahead and discard the negative value as the number of terms in a sequence cannot be negative so n comes out to be 16 let me get my problem back so this company made toys for 16 Helios such that these some of the toys made each year up to 16 years is 99 thousand