# Geometric Sequences

Geometric sequences, a fundamental concept in mathematics, unveil the beauty of patterns and order within numerical progressions. Defined by a constant ratio between successive terms, these sequences exhibit a distinctive multiplicative structure. Each term is derived by multiplying the preceding one by a fixed factor, resulting in a systematic progression. With applications ranging from finance to scientific modeling, understanding geometric sequences is crucial in various fields. This concise introduction aims to illuminate the core principles of geometric sequences, providing a gateway to their significance in mathematical analysis and practical problem-solving.

Questions

- How do you write the expression for the nth term of the geometric sequence #a_1=1000, r=1.005, n=60#?
- How do you write an nth term rule for #a_1=-1/2# and #a_4=-16#?
- How do you tell whether the sequence #100, 50, 25, 25/2, 25/4,...# is geometric?
- What would the formula for the nth term be given 0.3, -0.06, 0.012, -0.0024, 0.00048, ...?
- How do you find the common ratio of the geometric sequence 7,-14,28,-56?
- How do you find a general formula for each arithmetic sequence given 8th term is -20; 17th term is -47?
- How do you write a sequence that has three geometric means between 256 and 81?
- Whats the missing term in the geometric sequence 3/4, __, 3?
- What is the common ratio in this geometric sequence 4, 16, 64, 256, 1024, ...?
- Is the sequence -24, -12, 0 a geometric sequence?
- How do you find the next two terms of the geometric sequence #1.25, -1.5, 1.8,...#?
- What is the nth term of the geometric sequence 5,15,45,135?
- What is the 7th term of this geometric sequence 2, 6, 18, 54, …?
- How do you find a formula of the nth term if the 4th term in the geometric sequence is -192 and the 9th term is 196608?
- What is the 6th term of the geometric sequence 4, 20, 100,...?
- Use the geometric mean to find the 7th term in a geometric sequence if the 6th term is 8 and the 8th term is 18?
- How do you write the first five terms of the geometric sequence #a_1=3, r=sqrt5#?
- How do you write an explicit formula for this sequence: 3, 6, 12, 24?
- A geometric sequence has first term 54 and 4th term 2. What is the common ratio?
- For a geometric sequence, #t_7=5x+2# and #t_10=x - 23#. If the common ratio is 2, how do you find the value of t10?