SOLUTION: A concert room has spaces for 10 seats in the first row, 13 in the second, 16 in the third, and so on. a. If there are 20 rows in the theater, represent the total number of seat

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Question 1186524: A concert room has spaces for 10 seats in the first row, 13 in the second, 16 in the third, and so on.
a. If there are 20 rows in the theater, represent the total number of seats in the concert room in sigma notation.
b. What is the maximum number of seats that can be occupied in the said concert room?
Found 2 solutions by ikleyn, MathTherapy:
Answer by ikleyn(52788) About Me  (Show Source):
You can put this solution on YOUR website!
.
A concert room has spaces for 10 seats in the first row, 13 in the second, 16 in the third, and so on.
a. If there are 20 rows in the theater, represent the total number of seats in the concert room in sigma notation.
b. What is the maximum number of seats that can be occupied in the said concert room?
~~~~~~~~~~~~~~~~~~ 


(a)   The sequence is the arithmetic progression with the first term a = 10; the common difference d = 3

      and the number of terms n = 20.


      So, the formula for the n-th term of an arithmetic progression


          a%5Bn%5D = a%5B1%5D + (n-1)*d


      and it is the number of seats in the n-th row.


      The formula for the total number of seats is (sigma-notation)


          S = sum%28%2810+%2B+%28n-1%29%2A3%29%2C+n=1%2C20%29.



(b)  To answer this question, apply the formula for the sum of n terms of an arithmetic progression


         S%5Bn%5D = %28a%5B1%5D+%2B+%28%28n-1%29%2Ad%29%2F2%29%2An

         S%5B20%5D = %2810+%2B+%28%2820-1%29%2A3%29%2F2%29%2A20 = 770.

Solved.

---------------

For introductory lessons on arithmetic progressions see
    - Arithmetic progressions
    - The proofs of the formulas for arithmetic progressions
    - Problems on arithmetic progressions
    - Word problems on arithmetic progressions
in this site.

Also,  you have this free of charge online textbook in ALGEBRA-II in this site
    - ALGEBRA-II - YOUR ONLINE TEXTBOOK.

The referred lessons are the part of this online textbook under the topic "Arithmetic progressions".


Save the link to this textbook together with its description

Free of charge online textbook in ALGEBRA-II
https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson

into your archive and use when it is needed.



Answer by MathTherapy(10552) About Me  (Show Source):
You can put this solution on YOUR website!
A concert room has spaces for 10 seats in the first row, 13 in the second, 16 in the third, and so on.
a. If there are 20 rows in the theater, represent the total number of seats in the concert room in sigma notation.
b. What is the maximum number of seats that can be occupied in the said concert room?
Formula for the SUM of an AP: matrix%281%2C3%2C+S%5Bn%5D%2C+%22=%22%2C+%28n%2F2%29%282a%5B1%5D+%2B+%28n+-+1%29d%29%29, with matrix%283%2C3%2C+n%2C+%22=%22%2C+20%2C+a%5B1%5D%2C+%22=%22%2C+10%2C+d%2C+%22=%22%2C+3%29

 matrix%281%2C3%2C+S%5Bn%5D%2C+%22=%22%2C+%28n%2F2%29%282a%5B1%5D+%2B+%28n+-+1%29d%29%29 then becomes: 

Maximum number of seats with 20 rows, or