SOLUTION: A polygon of n side has {{{n(n-3)/2}}} diagonals. How many sides has a polygon with 77 diagonals? The sum of first n positive integers is {{{S=n(n+1)/2}}}. How many consecutive

Algebra ->  Test -> SOLUTION: A polygon of n side has {{{n(n-3)/2}}} diagonals. How many sides has a polygon with 77 diagonals? The sum of first n positive integers is {{{S=n(n+1)/2}}}. How many consecutive      Log On


   

Question 411529: A polygon of n side has n%28n-3%29%2F2 diagonals. How many sides has a polygon with 77 diagonals?
The sum of first n positive integers is S=n%28n%2B1%29%2F2. How many consecutive integers, starting with 1, are required to add up to 66?
Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!
a ploygon of n side has(n-3) diagonals. how many sides has a polygon with 77 diagonals.
the sum of first integers n is s=(n+1). find how many integers are required to add up to 66 

Both those formulas are wrong. A polygon with n sides has n%28n-3%29%2F2 diagonals,
not n-3. The sum of the first n integers n is S=%28n%28n%2B1%29%29%2F2, not n+1. 

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I will answer using the correct formulas, not the ones given.

How many sides has a polygon with 77 diagonals?

n%28n-3%29%2F2=77
n%28n-3%29=154
n%5E2-3n=154
n%5E2-3n-154=0
%28n%2B11%29%28n-14%29=0

n+11=0          n-14=0
   n= -11          n=14

We discard the negative answer.  The correct solution is a 14-sided
polygon.

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How many integers are required to add up to 66?

%28n%28n%2B1%29%29%2F2=66
n%28n%2B1%29=132 
n%5E2%2Bn=132
n%5E2%2Bn-132=0
%28n-11%29%28n%2B12%29=0

n-11=0       n+12=0
   n=11        n=-12

We discard the negative answer.  The correct solution is the first
11 integers:

1+2+3+4+5+6+7+8+9+10+11=66

Edwin