SOLUTION: It is the year 2040 and the Olympics are being held in the People’s Republic of Scotland. The turnout this year is expected to be larger than ever so the Scots have embarked upon

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 Click here to see ALL problems on Sequences-and-series Question 172792: It is the year 2040 and the Olympics are being held in the People’s Republic of Scotland. The turnout this year is expected to be larger than ever so the Scots have embarked upon the construction of the largest arena ever created by mankind. This arena will have stadium seating, thus each consecutive row of seating will have more seats than the last. The first row of seats has 300, the second has 350, the third has 400, the fourth has 450 and so on. If there are 700 rows of seating, how many people will the arena sit in total? Answer by jim_thompson5910(28717)   (Show Source): You can put this solution on YOUR website!In this problem (and in future problems), it is VERY critical to know the formula What this means is the sum from 1 to "n" is equal to . So something like First off, take note that this sequence is an arithmetic sequence (since the difference between each term is the same). So the formula for an arithmetic sequence is where "d" is the difference between two terms and is the first term. Since the first term is , this means that . Also, because each number is 50 more than the previous one, this means that (ie the difference is 50) So the formula for the sequence is where "n" starts at 0. To start at n=1, just subtract 50 from 300 to get (to shift the terms) So the formula we'll work with is where . So for instance, the third row has seats (which is what is given) Now plug in to find out how many seats are in the 700th row So there are 35,250 seats in the 700th row -------------------------------------------- So to find the sum of the seats, we'll add them like this: 300 + 350 + 400 + 450 + ... 35250 Now factor out the GCF 50 to get 50( 6 + 7 + 8 + 9 + ... 705 ) Now remember, the formula allows us to sum from 1 to "n". Since we're starting at 6, we need to add in AND subtract that same amount (to balance things out) like this: 50( 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + ... 705 - 1 - 2 - 3 - 4 - 5 ) So it turns out that 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + ... 705 = and So the expression 50( 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + ... 705 - 1 - 2 - 3 - 4 - 5 ) becomes 50( 248865 - 15 ) Subtract 50( 248850 ) Multiply 12,442,500 ============================================== Answer: So there are 12,442,500 seats in the stadium That's twelve million, four hundred forty two thousand, five hundred seats