w represents width
4w represents length
d represents diagonal
w2 + (4w)2 = d2
w2 + 16w2 = d2
17w2 = d2
±w√17 = d
The diagonal is the width times √17.
I believe it's 12 because 5*16=80. So, then you would subtract 4 from 16 and gat 12.
x = # of balcony seats
y = # of orchestra seats
We have to create a system of equations to solve this problem
x + y = 256
$8x + $12y = $2,716
We will solve this system of equations by elimination.
Multiply the first equation by -8
-8x - 8y = -2048
8x + 12y = 2716
Let's add the equations together
0 + 4y = 668
Simplify the left side
4y = 668
Divide both sides by 4
y = 167
We can subtract 167 from 257 to get the number of balcony seats.
257 - 167 = 90 balcony seats
There are 167 orchestra seats and 90 balcony seats
Answer:
The sum of the first 37 terms of the arithmetic sequence is 2997.
Step-by-step explanation:
Arithmetic sequence concepts:
The general rule of an arithmetic sequence is the following:
![a_{n+1} = a_{n} + d](https://tex.z-dn.net/?f=a_%7Bn%2B1%7D%20%3D%20a_%7Bn%7D%20%2B%20d)
In which d is the common diference between each term.
We can expand the general equation to find the nth term from the first, by the following equation:
![a_{n} = a_{1} + (n-1)*d](https://tex.z-dn.net/?f=a_%7Bn%7D%20%3D%20a_%7B1%7D%20%2B%20%28n-1%29%2Ad)
The sum of the first n terms of an arithmetic sequence is given by:
![S_{n} = \frac{n(a_{1} + a_{n})}{2}](https://tex.z-dn.net/?f=S_%7Bn%7D%20%3D%20%5Cfrac%7Bn%28a_%7B1%7D%20%2B%20a_%7Bn%7D%29%7D%7B2%7D)
In this question:
![a_{1} = -27, d = -21 - (-27) = -15 - (-21) = ... = 6](https://tex.z-dn.net/?f=a_%7B1%7D%20%3D%20-27%2C%20d%20%3D%20-21%20-%20%28-27%29%20%3D%20-15%20-%20%28-21%29%20%3D%20...%20%3D%206)
We want the sum of the first 37 terms, so we have to find ![a_{37}](https://tex.z-dn.net/?f=a_%7B37%7D)
![a_{n} = a_{1} + (n-1)*d](https://tex.z-dn.net/?f=a_%7Bn%7D%20%3D%20a_%7B1%7D%20%2B%20%28n-1%29%2Ad)
![a_{37} = a_{1} + (36)*d](https://tex.z-dn.net/?f=a_%7B37%7D%20%3D%20a_%7B1%7D%20%2B%20%2836%29%2Ad)
![a_{37} = -27 + 36*6](https://tex.z-dn.net/?f=a_%7B37%7D%20%3D%20-27%20%2B%2036%2A6)
![a_{37} = 189](https://tex.z-dn.net/?f=a_%7B37%7D%20%3D%20189)
Then
![S_{37} = \frac{37(-27 + 189)}{2} = 2997](https://tex.z-dn.net/?f=S_%7B37%7D%20%3D%20%5Cfrac%7B37%28-27%20%2B%20189%29%7D%7B2%7D%20%3D%202997)
The sum of the first 37 terms of the arithmetic sequence is 2997.