Question

You attend a comedy show in an auditorium. Each row of the auditorium contains 2 more seats than the row in front of it. There are 25 seats at the comedian's level. We refer to the comedian-level row as Row 0, the row behind Row 0 as Row 1, etc. Let PN denote the number of seats in row N.

(a) Find P1. Find P2.
(b) The sequence P0, P1, P2, P3, ... is sequence.
(c) How many seats are in Row 5?
(d) How many seats are in Row 40?
(e) Which is the first row with more than 90 seats? Row
(f) How many total seats are in Row 0 through Row 5?
(g) There are 41 rows in the auditorium (labeled Row 0 through Row 40).
How many seats are in the entire auditorium?

Answer 1

Explanation:

Given that Each row of the auditorium contains 2 more seats than the row in front of it. There are 25 seats at the comedian's level. We refer to the comedian-level row as Row 0, the row behind Row 0 as Row 1, etc. Let PN denote the number of seats in row N.

Thus no of seats in rows form an arithmetic progression with common difference =2

a = I term = P0 = 25

Hence general formula is
`P_n = 25+2n`

a) Find P1. Find P2.

27, 29

(b) The sequence P0, P1, P2, P3, ... is sequence.

Arithmetic

(c) How many seats are in Row 5?

`25+2(5) = 35`

(d) How many seats are in Row 40?

`25+40(2) = 105`

(e) Which is the first row with more than 90 seats?

`25+2n>90\\2n>65\\n>32.5`

Row 33

(f) How many total seats are in Row 0 through Row 5?

`25+27+...+35\\= (6)/(2) (25+35)\\= 180`

(g) There are 41 rows in the auditorium (labeled Row 0 through Row 40).

How many seats are in the entire auditorium?

Using Sum of 41 terms here

`(41)/(2) (25*2+40*2) = 2665`