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Let the first term be a and last term is 23.
The sum of the AM is 75.
(a+23)/2 x n = 75
(a+23) x n = 150

n needs to be a factor of 150. So n can be 2,3.5.6. n as 2 or 3 is not possible because of sum of 23 and any smaller one or two values less then 23 will not make 75.

So n can be 5 or 6.

If n is five then the last term (23+a)5 = 150. it boils down to a = 7

If a = 7, a + (n-1)d = 23
7 + 4d = 23
so d = 4

If we check with the above calculation the terms 23, 19, 15, 11, 7 satisfy the summation of 75. Hence n = 5 is the correct answer.
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ibakhronov
Sides of a polygon are in arithmetic progression that has a difference equal to 4. The perimeter of the polygon is 75 and the largest side is equal to 23. Find the number of sides of the polygon.
A. 5
B. 6
C. 7
D. 8
E. 9

The lengths are: 23, 19, 15 and so on, such that they add up to 75

Working with options is the easiest here since it's all about simply adding a few numbers.

For 5 sides:
23 + 19 + 15 + 11 + 7 = 15 * 5 = 75
(Note: Sum = Mean x number of terms; mean is the middle term)

(Clearly, we could have accommodated only one more side. Thereafter the sides would become negative. So, at the worst, if 5 sides didn't work, 6 must have worked)

Answer A


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