急!F6 Arithmetic Sequences 1q14

31.
(a)
Let a cm and d cm be the first term and the common difference of the arithmeticsequence.

AB, BC, CD, DE, ...... = a cm, a+d cm, a+2d cm, a+3d cm, ......

Length of ABCD :
a + (a + d) + (a + 2d) = 24
3a + 3d = 24
a + d = 8
a = 8 - d ...... [1]

The sum of area of P1, P2 and P3 :
(1/2)a² +(1/2)(a + d)² +(1/2)(a + 2d)² =105
a² +a² +2ad + d² +a² +4ad + 4d² =210
3a² +6ad + 5d² =210 ...... [2]

Put [1] into [2] :
3(8 - d)² +6(8 - d)d + 5d² =210
192 - 48d + 3d² + 48d - 6d² + 5d² = 210
2d² =18
d² =9
d = 3 or d = -3 (rejected)

Put d = 3 into [1] :
a = 8 - (3)
a = 5

The length of BC
= (a + d) cm
= (5 + 3) cm
= 8 cm

(b)
The difference of the lengths of AB and BC
= AB - BC
= -d cm
= -3 cm

(c)
The area of P10
= (1/2) * (a + 9d)² cm²
= (1/2) * (5 + 9*3)² cm²
= 512 cm²


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32.
(a)(i)
The perimeters (in cm) of A1, A2, A3 ...... forman arithmetic sequence :
the first term, a = 12, and the common difference, d = 4

The perimeter of A10
= (a + 9d) cm
= (12 + 9*4) cm
= 48 cm

(a)(ii)
Sum of nth terms ≤ 600
n [2a + (n - 1)d] / 2 ≤ 600
n [2*12 + (n - 1)*4] ≤ 1200
4n² + 20n ≤ 1200
n² + 5n ≤ 300
n² + 5n + 2.5² ≤ 300 + 2.5²
(n + 2.5)² ≤ 306.25
(n + 2.5) ≤ √306.25 for n > 0
n ≤ -2.5 + √306.25
n ≤ 15

Number of figures formed = 15

(b)(i)
(Area of A1) : (Area of A2) = 12² : (12 + 4)²
(3 cm²) : (Area of A2) = 144 : 256

Area of A2
= 3 x (256/144) cm²
= 16/3 cm²

(b)(ii)
(Area of A1) : (Area of A3) = 12² : (12 + 2*4)²
(3 cm²) : (Area of A2) = 144 : 400

Area of A3
= 3 x (400/144) cm²
= 25/3 cm²

(Area of A2) - (Area of A1)
= [(16/3) - 3] cm²
= 7/3 cm²

(Area of A3) - (Area of A2)
= [(25/3) - (16/3)] cm²
= 3 cm²

Since (Area of A2) - (Area of A1) ≠(Area of A3)- (Area of A2)
then the areas of A1, A2, A3, ...... DO NOT form an arithmetic sequence.