If o1, o2, on, angles formed around the point o and m(o1) = n degrees, m(o2) = n+1 degrees...m(on) = 2n-1 degrees , m(on + 1) = 2n degrees . Which is the value of n , natural number ?

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Since o1,o2,...,on are angles  around a point:

Then the sum of the measures of these angles is 360 degrees;

==> m(o1)+m(o2)+...+m(on)= 360

==> n + n+1 + n+2+ ...+2n-3+2n-2+2n-1 = 360

Rearrange:

(1+2+3+...+n-1 +n )+ n(n+1)=360

we know that:  1+2+3+...+n-1+n = n(n+1)/2

==> n(n+1)/2 +n(n+1)=360

n^2+n +2n^2+2n= 720

3n^2+3n=720

3n(n+1)=720

n(n+1)= 240

...

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Since o1,o2,...,on are angles  around a point:

Then the sum of the measures of these angles is 360 degrees;

==> m(o1)+m(o2)+...+m(on)= 360

==> n + n+1 + n+2+ ...+2n-3+2n-2+2n-1 = 360

Rearrange:

(1+2+3+...+n-1 +n )+ n(n+1)=360

we know that:  1+2+3+...+n-1+n = n(n+1)/2

==> n(n+1)/2 +n(n+1)=360

n^2+n +2n^2+2n= 720

3n^2+3n=720

3n(n+1)=720

n(n+1)= 240

n^2+n-240=0

(n+16)(n-15)=0

==> n =15 degrees.

(we will not consider n=-16 because it is a negative value and degrees should be positive values)

 

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