1. ## arithmetic sequences

A polygon has 10 sides. The lengths of the sides, starting with the smallest, form an arithmetic series. The perimeter of the polygon is 675 cm and the length of the longest side is twice that of the shortest side. Find, for this series:

(a) The common difference.

(b) The first term.

2. Originally Posted by Tweety
A polygon has 10 sides. The lengths of the sides, starting with the smallest, form an arithmetic series. The perimeter of the polygon is 675 cm and the length of the longest side is twice that of the shortest side. Find, for this series:

(a) The common difference.

(b) The first term.
Let $a$ be the shortest side and $d$ the difference so the sides are

$a, a+d, a+2d,\; \cdots \; a + 9d$

Adding gives $10a + 45d = 675\, (*)$, the condition that the longest is twice the shortes gives $2a = a + 9d$ or $a = 9d$. Use this in (*).

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### a polygon has 25 sides. the leght of which forms an arithmetic sequence

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