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1.
(i)
T(1) = 3(2)¹ = 6
T(2) = 3(2)² = 12
T(3) = 3(2)³ = 24
T(4) = 3(2)⁴ = 48
T(5) = 3(2)⁵ = 96

(ii)
T(1) = (-1)¹(3)⁰ = -1
T(2) = (-1)²(3)¹ = 3
T(3) = (-1)³(3)² = -9
T(4) = (-1)⁴(3)³ = 27
T(5) = (-1)⁵(3)⁴ = -81


2.
First term, a
common ratio, r = 2

T(5):
ar⁴ = 32
a(2)⁴ = 32
a = 2
First term, T(1) = 2


3.
(i)
T(1) = 3
T(2) = 6 = 3 x 2
T(3) = 12 = 3 x 2^2
T(4) = 24 = 3 x 2^3

T(n) = 3 x 2^(n - 1)

(ii)
T(1) = 0.2
T(2) = 0.6 = 0.2 x 3
T(3) = 1.8 = 0.2 x 3^2
T(4) = 5.4 = 0.2 x 3^3

T(n) = 0.2 x 3^(n - 1)


4.
First term, a = 1
Common ratio = 3/1 = 3

T(n):
a r^(n - 1) = 243
1 x (3)^(n - 1) = (3)^5
n - 1 = 5
n = 6

Number of terms = 6


5.
Let m be the geometric mean.

m/(-24) = (-6)/m
m² = 144
m = 12 or m = -12
The geometric mean is 12 or -12.


6.
T(2) and T(3) are the two geometric means.
First term, a = -0.5
Common ratio, r

T(4) = a r³
(-0.5) r³ = 4
r³ = -8
r = -2

T(2) = (-0.5) x (-2) = 1
T(3) = (-0.5) x (-2)² = -2

The two geometric means are 1 and -2.

7.
(i)
First term, a = 1
Common ratio, r = -2/1 = -2
No. of terms, n = 15

S(15)
= a [1 - r^n] / [1 - r]
= 1 x [1 - (-2)^15] / [1 - (-2)]
= (1 + 32768) / (1 + 2)
= 10923

(ii)
First term, a = 2
Common ratio, r = 6/2 = 3

T(n):
a r^(n - 1) = 4374
2 x (3)^(n - 1) = 4374
3^(n - 1) = 3^7
n - 1 = 7
n = 8

S(8)
= a [r^n - 1] / [r - 1]
= 2 x [3^8 - 1] / [3 - 1]
= 6560


8.
(i)
First term, a
Common ratio, r

T(2):
a r = 2 ...... [1]

T(6):
a r^5 = 1/8 ...... [2]

[2]/[1] :
r^4 = 1/16
r^4 = (1/2)^4
r = 1/2 or r = -1/2

When r = 1/2,[1]:
a (1/2) = 2
a = 4
General term = a r^(n - 1) = 4 x (1/2)^(n - 1)

When r = -1/2, [2]:
a (-1/2) = 2
a = -4
General term = a r^ (n - 1) = -4 x (-1/2)^(n - 1)

(ii)
When a = 4 and r = 1/2, Sum to infinity
= a / [1 - r]
= 4 / [1 - (1/2)]
= 8

When a = -4 and r = -1/2, Sum to infinity
= a / [1 - r]
= 4 / [1 - (-1/2)]
= 8/3