Question

A meeting hall has 20 seats in the first row, 24 seats in the second row, 28 seats in the third row, and so on and has in all 30 rows. How many seats are there in the meeting hall?

Answer 1

Because there are t₁ = 20 seats in the 1^st row, t₂ = 24 seats in the 2^nd row, t₃ = 28 seats in the 3^rd row and so on, the numbers of the seats in the rows are in an A.P.
Here,

First term, a = t₁ = 20
Common difference, d = t₂ – t₁ = 24 – 20 = 4
Now,
To find the total number of seats in all 30 rows of a hall, we need to find the sum of the first 30 terms of the A.P.
We know that the sum of n terms of an A.P is $S_n=\frac{n}{2}\left[2a+(n-1)d\right]$.
By taking a = 20, d = 4 and n = 30, we get:

$$\begin{gathered} S_{30}=\frac{30}{2}\left[2\times 20+(30-1)4\right] \\ =15\left[40+29\times 4\right] \\ =15\times 156 \\ =2340 \end{gathered}$$

Thus, there are 2340 seats in all 30 rows of the hall.