Chapter 1 Text work review

1. If it is a geometric sequence, state the common ratio r, and the general term t_n.

a) 1, 2, 4, 8, ...

   r = 2, t_n = 2^{n - 1}

b) 2, 4, 6, 8, ... not a geometric sequence

c) 3, -9, 27, -87, ...

   r = -3, t_n = 3(-3)^{n - 1}

d) 1, 1, 2, 4, 8, ... not a geometric sequence

e) 10, 15, 22.5, 33.75, ...

   r = 1.5, t_n = 10\( \cdot \)1.5^{n - 1}

f) -1, -5, -25, -125, ...

   r = 5, t_n = -5^n-1

3. Determine the first four terms:

a) t₁ = 2, r = 3

   2, 6, 18, 54

b) t₁ = -3, r = -4

   -3, 12, -48, 192

c) t₁ = 4, r = -3

   4, -12, 36, -108

d) t₁ = 2, r = 0.5

   2, 1, 0.5, 0.25

5. Determine the formula for the general term:

a) r = 2, t₁ = 3

   t_n = 3\( \cdot \)2^{n - 1}

b) 192, -48, 12, -3, ...

   r = -\( \frac{1}{4} \), t₁ = 192, t_n = 192(\( \frac{1}{4} \)^{n - 1}

c) t₃ = 5, t₆ = 135

   t₆ = t₃r³, r³ = \( \frac{135}{5} \) = 27 = 3³ ---> r = 3

   t₃ = t₁\( \cdot \)3² ---> t₁ = \( \frac{5}{9} \)

   t_n = \( \frac{5}{9} \)\( \cdot \)3^{n - 1}

d) t₁ = 4, t₁₃ = 16,384

   t₁₃ = 4r¹² = 16,384 ---> r¹² = 4096 = 2¹² ---> r = 2

   t_n = 4\( \cdot \)2^{n - 1} = 2²\( \cdot \)2^{n - 1} = 2^{n + 1}  = t_n