A small auditorium has 10 rows of seats. There are 12 seats in the first row and 16 seats in the second row. The number of seats in a row continues to increase by 4 with each additional row. What is the total number of seats in the auditorium?

352

156

300

252

Question

A small auditorium has 10 rows of seats. There are 12 seats in the first row and 16 seats in the second row. The number of seats in a row continues to increase by 4 with each additional row. What is the total number of seats in the auditorium?

352

156

300

252

rootbeer003 · Answer

@Hero

Hero · Answer

@Ultrilliam can probably answer this question

Ultrilliam · Answer

I'm busy ATM dealing with ASO and their poor decisions

Hero · Answer

@rootbeer003, how many questions do you have in total?

rootbeer003 · Answer

3

Hero · Answer

You have three more questions left?

rootbeer003 · Answer

including this one yes

Hero · Answer

Okay so this is what you do. We know there are ten rows of seats so we "count"

1st row --- 12 seats
2nd row --- 16 seats
3rd row ---  20 seats
...
nth row --- a_n seats
So we have to figure out a relationship between n and a_n.

We do this by starting with the first row and counting in the following manner:

1st row --- 12  + 0(4) seats
2nd row --- 12 +  1(4) seats
3rd row ---  12 +  2(4) seats
...
nth row --- 12 + (n -1)(4) seats

Hero · Answer

We use this pattern to figure out the number of seats in the 10th row.

Hero · Answer

10th row ---> 12 + (10 - 1)(4) seats

So @rootbeer003 how many seats are there in the 10th row?

rootbeer003 · Answer

48

Hero · Answer

Very good. Now we have to figure out the total number of seats in the auditorium. Do you have the sum formula with you?

rootbeer003 · Answer

no

Hero · Answer

$\sum_{\text{n = 1}}^{10} 12 + (n-1)(4)$

Hero · Answer

Ugh, latex is terrible with sums

rootbeer003 · Answer

idk how to use that formula..

Hero · Answer

@Ultrilliam you should fix this. LaTeX doesn't work well with Sum formulas at all

rootbeer003 · Answer

is it 300?

Hero · Answer

Are you guessing or did you actually use a formula to come up with that?

Hero · Answer

The other way to figure it out is to use the following formula:

$\text{Sum} = n\left(\dfrac{a_1 + a_n}{2}\right)$

rootbeer003 · Answer

What do i put in as a and n

Hero · Answer

In this case, $n = 10,  a_1 = 12, a_n = 48$,

$\text{Sum} = n\left(\dfrac{a_1 + a_n}{2}\right)$

Hero · Answer

In other words 
n represents total number of rows
$a_1$ represents the first term in the sequence
$a_n$ represents the last term in the sequence

Hero · Answer

The answer is 300, but you did not explain how you came up with it.

rootbeer003 · Answer

attachment

Hero · Answer

Post as a separate question please.

1 attachment