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?

Answer 1

(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)

Answer 2

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