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The interior angles of a convex polygon form an arithmetic progression with a common difference of 4°. Determine the number of sides of the polygon if its largest interior angle is 172°:-
Correct answer is '12'. Can you explain this answer?
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The interior angles of a convex polygon form an arithmetic progression...
Let's assume that the first angle of the arithmetic progression is a, and the common difference is d.
The sum of the interior angles of a convex polygon with n sides is given by the formula (n-2) * 180 degrees.
Since the interior angles form an arithmetic progression, we can write the sum of the angles as follows:
(n-2) * 180 = (a + (a+d) + (a+2d) + ... + (a+(n-3)d) + (a+(n-2)d))
To simplify this equation, we can use the formula for the sum of an arithmetic series:
(n-2) * 180 = (n/2) * (2a + (n-3)d)
Simplifying further:
360 - 360/n = 2a + (n-3)d
We know that the common difference is 4, so we substitute d = 4:
360 - 360/n = 2a + 4n - 12
Rearranging the equation:
2a + 4n - 12 = 360 - 360/n
2a = 360 - 360/n - 4n + 12
2a = 372 - 4n - 360/n
a = (372 - 4n - 360/n) / 2
Therefore, the first angle of the arithmetic progression is given by (372 - 4n - 360/n) / 2.
The sum of the interior angles of a convex polygon with n sides is given by the formula (n-2) * 180 degrees.
Since the interior angles form an arithmetic progression, we can write the sum of the angles as follows:
(n-2) * 180 = (a + (a+d) + (a+2d) + ... + (a+(n-3)d) + (a+(n-2)d))
To simplify this equation, we can use the formula for the sum of an arithmetic series:
(n-2) * 180 = (n/2) * (2a + (n-3)d)
Simplifying further:
360 - 360/n = 2a + (n-3)d
We know that the common difference is 4, so we substitute d = 4:
360 - 360/n = 2a + 4n - 12
Rearranging the equation:
2a + 4n - 12 = 360 - 360/n
2a = 360 - 360/n - 4n + 12
2a = 372 - 4n - 360/n
a = (372 - 4n - 360/n) / 2
Therefore, the first angle of the arithmetic progression is given by (372 - 4n - 360/n) / 2.
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The interior angles of a convex polygon form an arithmetic progression with a common difference of 4°. Determine the number of sides of the polygonif its largest interior angle is 172°:-Correct answer is '12'. Can you explain this answer?
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The interior angles of a convex polygon form an arithmetic progression with a common difference of 4°. Determine the number of sides of the polygonif its largest interior angle is 172°:-Correct answer is '12'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about The interior angles of a convex polygon form an arithmetic progression with a common difference of 4°. Determine the number of sides of the polygonif its largest interior angle is 172°:-Correct answer is '12'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The interior angles of a convex polygon form an arithmetic progression with a common difference of 4°. Determine the number of sides of the polygonif its largest interior angle is 172°:-Correct answer is '12'. Can you explain this answer?.
The interior angles of a convex polygon form an arithmetic progression with a common difference of 4°. Determine the number of sides of the polygonif its largest interior angle is 172°:-Correct answer is '12'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about The interior angles of a convex polygon form an arithmetic progression with a common difference of 4°. Determine the number of sides of the polygonif its largest interior angle is 172°:-Correct answer is '12'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The interior angles of a convex polygon form an arithmetic progression with a common difference of 4°. Determine the number of sides of the polygonif its largest interior angle is 172°:-Correct answer is '12'. Can you explain this answer?.
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