There is an auditorium with 27 rows of seats. There are 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 `15^(th)` row and also find how many total seats are there in the auditorium?

Answer

There are 48 seats in the 15th row. The total number of seats in the auditorium is 1242.

Explanation

20 seats in the first row, 22 seats in the second row, 24 seats in the third row,
`therefore 20, 22, 24, ...` is a sequence.
The number of seats increasing in consecutive rows
`=24-22=22-20=2`
`therefored=2`
There are 20 seats in the first row.
`thereforea=20`
`therefore20,22,24,`... is an A.P.
We have to find the number of seats in the 15th row. i.e. `t_(15)`.
`t_(n)=a+(n-1)d` ...(Formula)
`thereforet_(15)=20+(15-1)xx2`
`=20+14xx2`
`=20+28`
`=20+28`
Total number of seats in 27 rows, i.e. we have to find `S_(27)`
`S_(n)=(n)/(2)[2a+(n-1)d]` ...(Formula)
`thereforeS_(27)=(27)/(2)[2xx20+(27-1)xx2]` ...(Substituting the values)
`=(27)/(2)[40+26xx2]`
`=(27)/(2)[40+52]`
`=(27)/(2)xx92=27xx46`
`thereforeS_(27)=1242`