A sequence of figures is formed using tiles. Each tile is an equilateral triangle with side length 7 cm. The first figure consists of 1 tile. Each figure after the first is formed by adding 1 tile to the previous figure. How many tiles are used to form the figure in the sequence with perimeter 91 cm?
Solution
The first figure consists of one tile with perimeter . Each time an additional tile is added, the perimeter of the figure increases by 7 cm (one side length of a tile), because one side length of the previous figure is 'covered up' and two new side lengths of a tile are added to the perimeter for a net increase of one side length (or 7 cm). Since the first figure has perimeter 21 cm and we are looking for the figure with perimeter 91 cm, then the perimeter must increase by . Since the perimeter increases by 7 cm when each tile is added, then tiles need to be added to reach a perimeter of 91 cm. In total, this figure will have tiles.
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