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

Answer:

8

Step-by-step explanation:

Answer 2

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

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