Question

A theater has 30 seats in the first row, 33 seats in the second row, 36 seats in the third row, and so on in the same increasing pattern. If the
theater has 15 rows of seats, how many seats are in the theater?
A.765
B. 775
c. 810
D. 840

Answer 1

Answer:

A. 765

Explanation:

I couldn't really come up with a simpler way to solve this, but this is how I did it anyways:

I just kept counting by 3's and writing the numbers until I had 15 different numbers, then I added them all up in the calculator and got 765

Answer 2

The total number of seats in the theater are 765.  Hence the correct option is ‘A’.

SOLUTION:

It is given that a theater has 30 seats in the first row, 33 seats in the second row, 36 seats in the third row, and so on in the same increasing pattern.  

We also know that the theater has 15 rows of seats. We need to find out how many total seats are in the theater

We can put the given number of seats in a sequence as follows:

30,33,36,39,…..

By the above sequence we know that it follows the arithmetic progression.

The difference between the consecutive numbers is 3.

Firstly, we need to find the nth term which is the 15th term in this case. So that we can find the total number of seats in the theater

We can do so by the following formula of Arithmetic Progression:

$a_{n}=a_{1}+(n-1) d$

Where $a_n$ = the nth term of the sequence

$a_1$ =  the first term of the series.

n = the number of the term we are looking for

d = the difference between consecutive numbers.

So the 15th term is:

$\begin{aligned} a_{15} &=30+(15-1) 3 \\\\ a_{15} &=30+14 \times 3 \\\\ a_{15} &=30+42=72 \end{aligned}$

So, now we have to find the total number of seats in the theatre which can be done by the following formula:

$S_{n}=n \times \frac{a_{1}+a_{n}}{2}$

$\begin{array}{c}{S 15=15 \times \frac{30+72}{2}} \\\\ {S 15=15 \times\left(\frac{102}{2}\right)=765}\end{array}$

Therefore, the total number of seats in the theatre are 765. Hence the correct option is ‘A’.