```Question 172792
In this problem (and in future problems), it is VERY critical to know the formula

{{{sum(i,i=1,n)=(n(n+1))/2}}}

What this means is the sum from 1 to "n" is equal to {{{(n(n+1))/2}}}. So something like {{{1+2+3+4+5 = (5(5+1))/2 = 15}}}

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

{{{a[n]=dn+a[1]}}} where "d" is the difference between two terms and {{{a[1]}}} is the first term.

Since the first term is {{{300}}}, this means that {{{a[1]=300}}}. Also, because each number is 50 more than the previous one, this means that {{{d=50}}} (ie the difference is 50)

So the formula for the sequence is {{{a[n]=50n+300}}} where "n" starts at 0. To start at n=1, just subtract 50 from 300 to get {{{a[n]=50n+250}}} (to shift the terms)

So the formula we'll work with is {{{a[n]=50n+250}}} where {{{n>=1}}}. So for instance, the third row has {{{50(3)+250=150+250=400}}} seats (which is what is given)

Now plug in {{{n=700}}} to find out how many seats are in the 700th row

{{{a[700]=50(700)+250=35000+250=35250}}}

So there are 35,250 seats in the 700th row

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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 {{{sum(i,i=1,n)=(n(n+1))/2}}} allows us to sum from 1 to "n". Since we're starting at 6, we need to add in {{{1+2+3+4+5}}} AND subtract that same amount (to balance things out) like this:

50( <font color=red>1 +  2 + 3 + 4 + 5</font> + 6 + 7 + 8 + 9 + ... 705 <font color=red>- 1 - 2 - 3 - 4 - 5</font> )

So it turns out that

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + ... 705 = {{{sum(i,i=1,705)=(705(705+1))/2 = (705(706))/2 = 705(353) = 248865}}}

and {{{- 1 - 2 - 3 - 4 - 5=-15}}}

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

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