Question

There is an auditorium with 27 rows of seats. There are 20 seats in the first row, 22 seats in the second row, 24 seats in the third row and so on. Find the number of seats in the 15^th row and also find how many total seats are there in the auditorium ?

Answer 1

Total number of rows = 27
Number of seats in first row = 20
Number of seats in second row = 22
Number of seats in third row = 24

Total number of seats = 20 + 22 + 24 + ..... + upto 27 rows

Here,
a = 20
d = 2
n = 27

Now,
$$\begin{gathered} S_n=\frac{n}{2}\left(2a+\left(n-1\right)d\right) \\ {S}_{27}=\frac{27}{2}\left(2a+\left(27-1\right)d\right) \\ =\frac{27}{2}\left(2\left(20\right)+26\left(2\right)\right) \\ =\frac{27}{2}\left(40+52\right) \\ =\frac{27}{2}\left(92\right) \\ =27\times 46 \\ =1242 \end{gathered}$$

Hence, there are 1242 seats in the auditorium.

Also,
$$\begin{gathered} a_{15}=a+\left(15-1\right)d \\ =20+14\left(2\right) \\ =20+28 \\ =48 \end{gathered}$$

Hence, the number of seats in the 15^th row is 48.