Question

The interior angles of a n’sided polygon are in AP. The smallest angle and the common difference of
the AP are both real numbers greater than or equal to 24. What is the maximum values of n possible?​

Answer 1

Answer:

n=12

Step-by-step explanation:

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Answer 2

The maximum possible value of n is 7

Step-by-step explanation:

Let the first term of the AP be a and common difference be d

Then

nth term of the AP

$a_n=a+(n-1)d$

Sum of the n terms of AP

$S_n=\frac{n}{2}[2a+(n-1)d]$

If these n terms of the AP constitute n interior angles of a polygon

Then

The largest angle of the polygon should be less than $180^\circ$

i.e. $a+(n-1)d<180^\circ$

$\implies n<\frac{180^\circ-a}{d}+1$

For minimum value of a and d = 24

$n<\frac{180-24}{24}+1$

$\implies n<7.5$

$\implies n\le7$  

Therefore, maximum side of the polygon is 7

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