Question

The number of seats in each row of a theater forms an arithmetic sequence. The fifth row contains 22 seats. The tenth row contains 37 seats . How many seats are in the first row?

Answer 1

Let
`s_n` denote the number of seats in the
`n`-th row.
`s_n` is arithmetic, so

`s_n=s_(n-1)+d`

for some constant
`d`.

We're told
`s_5=22` and
`s_(10)=37`, so that

`s_(10)=s_9+d`

`s_(10)=(s_8+d)+d=s_8+2d`

`s_(10)=(s_7+d)+2d=s_7+3d`

and so on up to

`s_(10)=s_5+5d\implies37=22+5d\implies5d=15\implies d=3`

The pattern continues:

`s_5=s_4+3`

`s_5=(s_3+3)+3=s_3+2\cdot3`

and so on up to

`s_5=s_1+4\cdot3\implies22=s_1+12\implies\boxed{s_1=10}`

Answer 2

8

Explanation: