Question
The first row of a concert hall has 25 seats, and each row after the first has one more seat than the
row before it. There are 32 rows of seats. 35 students from a class want to sit in the same row.
a. In which row would the class sit?
b. How many seats are there in the concert hall?
row before it. There are 32 rows of seats. 35 students from a class want to sit in the same row.
a. In which row would the class sit?
b. How many seats are there in the concert hall?
Answers
oobleck
(a) row n has 25+(n-1)*1 = n+24 seats
so row 11 has 35 seats
(b) S32 = 32/2 (2*25 + 31*1)
so row 11 has 35 seats
(b) S32 = 32/2 (2*25 + 31*1)
Bosnian
The number of seats in rows is an arithmetic progression.
For an arithmetic progression:
an = a + ( n - 1 ) d
where
a = the initial term
d = the common difference of successive members
an = the nth term
In this case:
a = 25 , d = 1
an = 25 + ( n - 1 ) ∙ 1
an = 25 + n - 1
an = 24 + n
a.
35 = 24 + n
Subtract 24 to both sides
35 - 24 = n
11 = n
n = 11
b.
Sum of the n terms of an AP:
S = n [ 2 a + ( n - 1 ) d ] / 2
In this case:
n = 32 , a = 25 , d = 1
S = n [ 2 a + ( n - 1 ) d ] / 2
S = 32 ∙ [ 2 ∙ 25 + ( 32 - 1 ) ∙ 1 ] / 2
S = 32 ∙ ( 50 + 31 ) / 2
S = 32 ∙ 81 / 2
S = 2592 / 2
S = 1296
For an arithmetic progression:
an = a + ( n - 1 ) d
where
a = the initial term
d = the common difference of successive members
an = the nth term
In this case:
a = 25 , d = 1
an = 25 + ( n - 1 ) ∙ 1
an = 25 + n - 1
an = 24 + n
a.
35 = 24 + n
Subtract 24 to both sides
35 - 24 = n
11 = n
n = 11
b.
Sum of the n terms of an AP:
S = n [ 2 a + ( n - 1 ) d ] / 2
In this case:
n = 32 , a = 25 , d = 1
S = n [ 2 a + ( n - 1 ) d ] / 2
S = 32 ∙ [ 2 ∙ 25 + ( 32 - 1 ) ∙ 1 ] / 2
S = 32 ∙ ( 50 + 31 ) / 2
S = 32 ∙ 81 / 2
S = 2592 / 2
S = 1296