Question 411529
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

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Both those formulas are wrong. A polygon with n sides has {{{n(n-3)/2}}} diagonals,
not n-3. The sum of the first n integers n is {{{S=(n(n+1))/2}}}, 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(n-3)/2=77}}}
{{{n(n-3)=154}}}
{{{n^2-3n=154}}}
{{{n^2-3n-154=0}}}
{{{(n+11)(n-14)=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?

{{{(n(n+1))/2=66}}}
{{{n(n+1)=132}}} 
{{{n^2+n=132}}}
{{{n^2+n-132=0}}}
{{{(n-11)(n+12)=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</pre>