"In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio."
A sequence of numbers where each term is the product of the previous term and a constant ratio.
"For example, the sequence 2, 6, 18, 54, ... is a geometric progression with common ratio 3. Similarly, 10, 5, 2.5, 1.25, ... is a geometric sequence with common ratio 1/2."
"The common ratio is a fixed, non-zero number that is used to find each term after the first by multiplying the previous one."
"The sequence 2, 6, 18, 54, ... is a geometric progression with common ratio 3."
"Yes, a geometric sequence can have fractions as the common ratio. For instance, 10, 5, 2.5, 1.25, ... is a geometric sequence with a common ratio of 1/2."
"Yes, examples of a geometric sequence are powers rk of a fixed non-zero number r, such as 2k and 3k."
"The general form of a geometric sequence is a, ar, ar^2, ar^3, ar^4, ..."
"The common ratio must be a non-zero number (r ≠ 0), and the scale factor must be a non-zero start value (a ≠ 0)."
"The sum of a geometric progression's terms is called a geometric series."
"The common ratio determines the relationship between each term and the previous term, as it is multiplied to find the next term in the sequence."
"If the common ratio in a geometric progression is 1, the sequence becomes a constant sequence."
"Yes, the first term in a geometric sequence plays a crucial role in determining subsequent terms through the multiplication with the common ratio."
"Yes, the common ratio in a geometric progression can be negative, as long as it is a fixed, non-zero number."
"A non-zero common ratio ensures that each term in the geometric progression is obtained by multiplying the previous term, leading to a pattern of growth or decay."
"No, not all non-zero sequences are geometric progressions. In a geometric progression, there is a consistent multiplication relationship between each term and the previous one."
"Yes, geometric progressions are also known as geometric sequences."
"For example, the sequence 10, 5, 2.5, 1.25, ... is a geometric sequence with a common ratio of 1/2."
"As the common ratio in a geometric progression increases, the terms in the sequence become larger at a faster rate."
"Yes, a geometric progression can have a common ratio greater than 1. This leads to exponential growth in the terms of the sequence."
"No, the scale factor in a geometric sequence cannot be equal to zero. It must have a non-zero start value."