Final answer:
The terms of a geometric sequence do not always increase. Whether they increase or decrease depends on whether the common ratio is greater than, less than, equal to 1, or negative.
Step-by-step explanation:
The statement that 'The terms of a geometric sequence always increase' is false. In a geometric sequence, each term is found by multiplying the previous term by a constant called the common ratio. If the common ratio is greater than 1, the terms will increase; if the common ratio is between 0 and 1, the terms will decrease; and if the common ratio is negative, the terms will alternate between positive and negative values, thus not always increasing. For example, the geometric sequence 2, 4, 8, 16,... has a common ratio of 2, and the terms increase.
However, the sequence 10, 5, 2.5, 1.25,... has a common ratio of 0.5, and the terms decrease. The statement in the question, 'The terms of a geometric sequence always increase' is false. A geometric sequence is a sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. This means that the terms can either increase or decrease, depending on the common ratio. For example, consider the geometric sequence with a common ratio of 2. The terms of the sequence would be: 1, 2, 4, 8, 16,... In this case, the terms are increasing. However, if we consider a geometric sequence with a common ratio of -2, the terms would be: 1, -2, 4, -8, 16,... Now, the terms are decreasing.