F.6 math arithmetic sequence~~－hzb53jl55v的部落格

標題:

F.6 math arithmetic sequence~~

發問:

There are 60 similar figures. The perimeter of the 1st figure is 5 cm. The perimeter of each succeeding figure is 2 cm longer than that of the previous one.It is known that the area of the 1st figure is 2cm^2.(a) Find the areas of the 2nd, 4th and 6th figures each.(b) Do the areas of the 2nd, 4th,... 顯示更多 There are 60 similar figures. The perimeter of the 1st figure is 5 cm. The perimeter of each succeeding figure is 2 cm longer than that of the previous one. It is known that the area of the 1st figure is 2cm^2. (a) Find the areas of the 2nd, 4th and 6th figures each. (b) Do the areas of the 2nd, 4th, 6th,...,60th figures from an arithmetic sequence? Explain briefly.

最佳解答:

There are 60 similar figures. The perimeter of the 1st figure is 5 cm.The perimeter of each succeeding figure is 2 cm longer than that of theprevious one. It is known that the area of the 1st figure is 2cm2. (a) Find the areas of the 2nd, 4th and 6th figures each. (b) Do the areas of the 2nd, 4th, 6th,...,60th figures from an arithmeticsequence? Explain briefly. Solution : (a) T(n) cm : The perimeter of the nth figure, where T(1) = 5 Common difference = 2 T(2) = 5 + (2 - 1)x2 = 7 T(4) = 5 + (4 - 1)x2 = 11 T(6) = 5 + (6 - 1)x2 = 15 ratio of areas of two similar figures it equal to the ratio of the squares ofperimeters. (Area of the 1st figure) : (Area of the nth figure) = (Perimeter of the 1st figure)2 : (Perimeter of the nthfigure)2 (2 cm2) : (Area of the 2nd figure) = (5 cm)2 : (7 cm)2 Area of the 2nd figure = 2 x 72 / 52 = 3.92 cm2 (2 cm2) : (Area of the 4th figure) = (5 cm)2 : (11 cm)2 Area of the 4th figure = 2 x 112 / 52 = 9.68 cm2 (2 cm2) : (Area of the 6th figure) = (5 cm)2 : (15 cm)2 Area of the 6th figure = 2 x 152 / 52 = 18 cm2 (b) (Area of the 4th figure) - (Area of the 2nd figure) = (9.68 - 3.92)2 = 5.76 cm2 (Area of the 6th figure) - (Area of the 4th figure) = (18 - 9.68)2 = 8.32 cm2 The differences are not equal, thus the areas of the 2nd, 4th, 6th,...,60th figures do not from anarithmetic sequence.