Question

An auditorium has 15 rows with 10 seats in the first row, 12 in the second row, 14 in the third row, and so forth. How many seats are in the auditorium?

Answer 1

there are 360 seats in the auditorium 

Answer 2

Answer: There are total 360 seats in the auditorium.

Step-by-step explanation: Given that an auditorium had 15 rows with 10 seats in the first row, 12 in the second row, 14 in the third row, and so forth.

We are to find the total number of seats in the auditorium.

Let us write the number of seats in the consecutive rows as a sequence as follows :

10, 12, 14, . . .

If a(n) denote the number of seats in the n-th row, then we see that

`a(2)-a(1)=12-10=2,\\\\a(3)-a(2)=14-12=2,\\\\\vdots~~~\vdots`

So, there is a common difference of 2 in the consecutive terms of the given sequence. That is, the sequence is an arithmetic one.

The total number of seats will be equal to the sum of the first 15 terms of the sequence.

We know that

the sum of first n terms of an ARITHMETIC sequence with first term a and common difference d is given by

`S_n=(n)/(2)(2a+(n-1)d).`

In our sequence, a = 10 and d = 2.

So, the sum of first 15 terms will be

`S_15=(15)/(2)(2*10+(15-1)2)=(15)/(2)(20+28)=(15)/(2)*48=360.`

Thus, there are total 360 seats in the auditorium.