急 F6 Geometric Sequences 02q15

33.
(a)(i)
The first term, a = 2
The common ratio, r = 2

The sum of the first n terms, Sn
= a (rⁿ - 1) / (r - 1)
= 2 (2ⁿ - 1) / (2 - 1)
= 2ⁿ⁺¹ - 2

(a)(ii)
The sum of the first 2n terms, S2n
= a (r²ⁿ - 1) / (r - 1)
= 2 (2²ⁿ - 1) / (2 - 1)
= 2²ⁿ⁺¹ - 2

(b)(i)
S2n = 65 Sn
2²ⁿ⁺¹ - 2 = 65(2ⁿ⁺¹ - 2)
2x2²ⁿ - 2 = 65(2x2ⁿ) - 2x65
2²ⁿ - 1 = 65(2ⁿ) - 65
2²ⁿ - 65(2ⁿ) + 64 = 0

(b)(ii)
2²ⁿ - 65(2ⁿ) + 64 = 0

Let u = 2ⁿ
u² - 65u + 64 = 0
(u - 1)(u - 64) = 0
u = 1 or u = 64
2ⁿ = 2⁰ or 2ⁿ = 2⁶
n = 0 (rejected) or n = 6


34.
(a)(i)
Let d the common difference.
Then, a = b - d and c = b + d

a² + b² = c² (Pythagorean theorem)
b² + (b - d)² = (b + d)²
b² + b² - 2bd + d² = b² + 2bd + d²
b² - 4bd = 0
b(b - 4d) = 0
b = 0 (rejected) or b = 4d

When common difference, d = 1 :
a = 3, b = 4, c = 5

When common difference, d = 2 :
a = 6, b = 8, c = 10

When common difference, d = 3 :
a = 9, b = 12, c = 15

(a)(ii)
a : b : c = 3 : 4 : 5

(b)(i)
Let r be the common ratio.
Then, b = ar and b = ar²

a² + b² = c² (Pythagorean theorem)
a² + (ar)² = (ar²)²
1 + r² = r⁴
(r²)² - r² - 1 = 0
r² = (1 + √5)/2 or r² = (1 - √5)/2 (rejected)
r = √[(1 + √5)/2] or r = -√[(1 + √5)/2] (rejected)

Hence, r = 1.272 and r² = 1.618

When a = 1 : b = 1.272, c = 1.618
The common ratio, r = 1.272

When a = 2 : b = 2.544, c = 3.236
The common ratio, r = 1.272

When a = 3 : b = 3.816, c = 4.854
The common ratio, r = 1.272

(b)(ii)
a : b : c = 1 : 1.272 : 1.618