At a faculty costume party, each teacher from transylvania Middle School came dressed as his or her favorite creepy creature. The teachers arrived one at a time, and when a new teacher came in, each of the teachers already there lined up at the door and shook hands with the newcomer. Count Dracula(Mr.Blab) kept track of the number of hand shkes exchanged. He said that there were 171 handshakes exchanged at the party.

How many teachers came to the costume party

the first teacher did not shake his own hand.

so, lets see how many handsakes Read in two lines,number people,and totalhandshakes.

2 3 4 5 6 7 8 9 10 11 12 13 14
1 3 6 10 15 21 28 36 45 55 66 78 91

15 16 17 18 19
105 120 136 153 171

19

Someone plz tell me the answer

To solve this problem, we can use a mathematical formula to determine the number of teachers who attended the costume party.

Let's assume that the number of teachers who attended the party is represented by 'n'. Each teacher who arrives shakes hands with all the teachers already present. So, the first teacher who arrives does not shake hands with anyone, the second teacher shakes hands with one person, the third teacher shakes hands with two people, and so on.

If we denote the number of handshakes made by a teacher who arrived at position 'x' as 'h(x)', we can see a pattern:

h(1) = 0
h(2) = 1
h(3) = 2
...
h(n) = (n-1)

To find the total number of handshakes made, we can use the formula:

Total number of handshakes = h(1) + h(2) + h(3) + ... + h(n)

In this case, the total number of handshakes is given as 171, so we have:

171 = h(1) + h(2) + h(3) + ... + h(n)

Now, we need to find the value of 'n' that satisfies this equation.

Let's calculate the sum h(1) + h(2) + h(3) + ... + h(n) for some values of 'n':

For n = 1, h(1) = 0
For n = 2, h(1) + h(2) = 0 + 1 = 1
For n = 3, h(1) + h(2) + h(3) = 0 + 1 + 2 = 3
For n = 4, h(1) + h(2) + h(3) + h(4) = 0 + 1 + 2 + 3 = 6

From these calculations, we can see that the sum h(1) + h(2) + h(3) + ... + h(n) is the sum of the first n-1 positive integers, which is given by the formula:

Sum of first n-1 positive integers = (n-1)(n-1+1)/2 = n(n-1)/2

So now, we can rewrite the equation as:

171 = n(n-1)/2

To solve for 'n', we can rearrange the equation:

n(n-1)/2 = 171
n(n-1) = 342

To find the value of 'n', we can iterate through the possible values of 'n' until we find two consecutive integers whose product is 342. By doing this, we can determine that the value of 'n' is 19.

Therefore, 19 teachers came to the costume party.