## taylorx 2 years ago ALGEBRA 2 HELP PLEASE WILL FAN AND MEDAL. In a 20-row theatre, the number of seats in a row increases by three with each successive row. The first row has 18 seats. a. write an arithmetic series to represent the number of seats in a theatre b. find the total seating capacity of the theater. c. front row tickets for a concert cost 60$. after every 5 rows, the ticket price goes down by 5$. what is the total amount of money generated by a full house? (show work please)

1. tafkas77

Do you understand this topic at all, or are you completely lost?

2. taylorx

@tafkas77 i'm completely lost. :(

3. tafkas77

That's perfectly fine. :) Just so you know, you don't have to fan me. I don't mind helping when I can. :) Give me just a moment, okay?

4. tafkas77

I would love to stay and help you, but something has just come to my attention. Here is an example that I worked out for you, okay? Arithmetic sequences allow you to predict the next number in a sequence by defining the pattern. If that doesn't make sense, I can give an example in a random sequence: 2, 6, 14, Here's the equation for this sequence: a_n +1 = 2a_n + 2 a_2 = 2(2) + 2; which equals 6. That's the second term. a_3 = 2(6) + 2; which equals 14. That's the third term. As you can see, you ALWAYS add the number you got from the previous equation to the next equation. This same thing can be applied to your question, okay? we know that a_n + 3 = the amount of seats in the room (or S) For the first row, we could look at it like this: a_n + 3 = 18 If you finish this equation you'll find that a_n = 15. :)

5. tafkas77

If you continue on with this equation, you can find out the next row, and the next row, and the next row.... :D Hope that helped!

6. tafkas77

Okay. Where were you lost? Or - from 1 - 10, 1 being completely lost and 10 being confident, how do you feel about this question?

7. tafkas77

@taylorx

8. taylorx

1 @tafkas77 . can you just help me through each step>? that'd be great

9. tafkas77

10. taylorx

@tafkas77 okay:)

11. tafkas77

I don't think I am equipped to help you on this one. I'm sorry. :/ @amistre64

12. amistre64

wow, taylor and tafkas look soooo much alike its hard to tell which post is which :)

13. amistre64

In a 20-row theatre, there are going to be rows 1 thru 20 the number of seats in a row increases by three with each successive row. each new row has 3 more seat than the one before it The first row has 18 seats. we know the first row has 18 seats row 1 has 18 seats row 2 has: 18+3 seats row 3 has: 18+3 + 3 seats row 4 has: 18+3 +3 +3 seats, this can start to be written as row 1 has 18 + 3(0) seats row 2 has: 18 + 3(1) seats row 3 has: 18 + 3(2) seats row 4 has: 18 + 3(4) seats does this make sense to you?

14. amistre64

row 4 has: 18 + 3(3) seats ... typoed it

15. amistre64

row 1 has 18 + 3(0) seats row 2 has: 18 + 3(1) seats row 3 has: 18 + 3(2) seats row 4 has: 18 + 3(3) seats ^^ ^^^ this value compares to this value notice that the nth row has 3(n-1) seats, therefore $R_n=18+3(n-1)~seats$

16. amistre64

when we know what produces the sequence of seats per row, we can then create a series, which is just a summation of the sequence:$\sum_{n=1}^{k}18+3(n-1)$ or simplified to $\sum_{n=1}^{k}15+3n$

17. taylorx

so would rn=18+3(n-1) seats be the arithemetic series @amistre64

18. amistre64

no, the vocabulary for this type of stuff is: a sequence represents a recurrsive pattern: 18, 21, 24, 27, 30, ... this pattern can be generated by the recurrsive equation: 15+3n, for simplicities sake a series is the sum of a sequence: 18 + 21 + 24 + 27 + 30 + ...

19. amistre64

a series keeps a running tab, a cumulative account of what has happened

20. taylorx

so how do you write the arithmetic series ? @amistre64

21. amistre64

its just the sum of the recursive equation:$\sum_{n=1}^{20}15+3n$

22. taylorx

@amistre64 thank you. so how do you find the seating capacity, thank you.

23. amistre64

the total number of seats can be worked out by hand, or by mathing techniques

24. amistre64

notice that the summation proceeds as: row1 + row 2 + row 3 + row 4 ... +row 20 --------- total number of seats

25. amistre64

row1: 15+3(1) row2: 15+3(2) row3: 15+3(3) row4: 15+3(4) .... row20: 15+3(20) ----------------- 15(20) + 3(1+2+3+4+...+20)

26. taylorx

@amistre64 thank you so much! now all i need is : front row tickets for a concert cost 60$. after every 5 rows, the ticket price goes down by 5$. what is the total amount of money generated by a full house?

27. amistre64

so, this tells us that we simply need to take the sum of the first 5 rows, and times it by 60 add that to the sum of the last 15 rows, times it by 5

28. amistre64

$65\sum_{1}^{5}(15+3n)+5\sum_{6}^{20}(15+3n)$ $65(15)(5)(3)\sum_{1}^{5}n+5(15)(15)(3)\sum_{6}^{20}n$

29. taylorx

@amistre64 is there anyway to simplify that.

30. taylorx

and @amistre64 what is the total money generated by a full house?

31. amistre64

not really, but its not that bad if you work it out in steps

32. amistre64

how many seats are in the first 5 rows? times 65 plus, how many seats are in the last 15 rows? times that by 5 add up the results

33. taylorx

okay thank you very much:) @amistre64

34. amistre64

try it out, and if you need me to dbl chk your results, i can do that :) i just want to steer clear of just handing over an answer is all

35. tafkas77

@amistre64 I love your methods. Lifesaver! :D Thank you so much! You helped both of us! :)

36. taylorx

@amistre64 for the recursive equation that you gave me on step a, is it supposed to be written with "k" on top or "20" on top , because you wrote it as both. thanks

37. amistre64

20 is more suited to the problem, im just forgot to make the general case 100% suited to your situation :) but good call on that

38. taylorx

@amistre64 thanks :) you are such a lifesaver

39. amistre64

:) might have to bail out soon, my kids are causing issues at school :/

40. taylorx

@amistre64 sorry to hear that, come back soon :)

41. amistre64

ill be back tomorrow ;) good luck