3. 1.3 Geometric Sequences

3.1. 1.3 Extra Practice

1.  Is each sequence geometric? If it is, state the common ratio and a formula to determine the general term in the form tn = t1rn – 1.

a) 11, 33, 99, 297, …

b) 6, 12, 18, 24, …

c) \( \frac{1}{3}, \frac{2}{3}, \frac{4}{3}, \frac{8}{3}, ... \)

d) 0.5, 0.2, 0.08, 0.032, …

2.  Write the first four terms of each geometric sequence.

a) t1 = 7, r = –3

b) t1 = –8, r = \( \frac{1}{2} \)

c) tn = 3(0.6)n – 1

d) tn = (–4)n

3.  Determine the number of terms in each geometric sequence.

a) 4, 12, 36, ..., 78 732

b) 5\( \sqrt[]{2} \), 10, 10\( \sqrt[]{2} \), ..., 640

c) t1 = 5, r = \( - \frac{1}{2} \), tn = \( \frac{5}{64} \)

d) t1 = \( \frac{1}{4} \), r = 3,  tn = 44 286.75

4.  Determine the nth term of each geometric sequence.

a) t1 = 2, r = 7

b) 6, –18, 54, –164, …

c) t1 = 7, t5 = 1792

d) r = \( \frac{1}{4} \), t8 = \( \frac{1}{4} \)

5.  Determine the unknown terms in each geometric sequence.

a) 18, ___, ___, 6174

b) ___, 4, ___, ___, 108

c) 5, ___, ___, ___, 80

6.  The first term of a geometric sequence is 0.1; the tenth term is 26 214.4. Determine the value of the common ratio.

7.  Determine the first term, the common ratio, and an expression for the general term of each geometric sequence.

a) t5 = 900,  t7 = 0.09

b) t3 = –1728,  t6 = 373 248

c) t5 = 28,  t11 = 1792

d) t2 = 3,  t4 = 0.75

8.  The following sequences are geometric. What is the value of each variable?

a) 8x – 12, 16, 64, 256, …

b) 25, 5, 1, 2y – 1, …

9.  For a geometric sequence t4 = 4x + 8 and t7 = x – 4. If the common ratio is \( \frac{1}{2} \), what is the first term?

10.  An excavating company has a digger that was purchased for $240 000. It is depreciating at 12% per year.

a) Determine the next three terms of this geometric sequence.

b) Determine the general term. Define your variables.

c) How much will the digger be worth in 7 years?

d) How long will it take before the equipment is worth less than $120 000?