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Mathematics 13 Online
OpenStudy (anonymous):

PLEASE HELP!!!!!!! I'm in 7th grade and have to find an expression for the following: Door bell rings, 1 guest there, door bell rings again, 2 more than last time, door bell rings again, 2 more than last time, and so on. I have to determine how many guests would be there after 20th doorbell. I know the answer is 400, but need the algebraic expression. Can anyone help?

OpenStudy (anonymous):

\[ a_1=1\\ a_n=a_{n-1} +2 \]

OpenStudy (anonymous):

You need \[ a_1+ a_2 + \cdots + a_{20}=400 \]

OpenStudy (anonymous):

So 1st bell: 1 guest 2nd bell: 3 guests 3rd bell: 5 gueses ... so on until 20th bell: 1+2(20-1)=39 Are you familiar with arithmetic series? If yes, then the algebraic expression is \[\sum_{n=1}^{20}2n-1=\frac{20}{2} \times (1+39)=400\] If not, we can try a different way.

OpenStudy (anonymous):

Not familiar with that, but i get the first part.

OpenStudy (anonymous):

OK. So let's try a more graphical way. So you have to find the sum 1+3+5+...+37+39 So double it you have 1+3+5+...+37+39+1+3+5+...+37+39=2(1+3+5+...+37+39) |dw:1355804236569:dw| Excuse my bad drawing but you can see that every single term of the above sum add to 40 Since there are 20 of the 40's added together (because there are 20 bells, remember?) so the picture above add up to 40*20=800. Remember that we double the whole sum, so we need to divide the thing by 2, which yields answer 400. So 1+3+5+...+37+39=400

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