In a geometric sequence, what does r represent?

In a geometric sequence, ( r ) represents the common ratio. This is a fundamental characteristic of geometric sequences, which are defined by the property that each term after the first is found by multiplying the previous term by a constant factor, known as the common ratio.

For example, in a geometric sequence like 2, 6, 18, 54, the common ratio can be calculated by dividing any term by the preceding term. In this case, ( 6 \div 2 = 3 ), ( 18 \div 6 = 3 ), and ( 54 \div 18 = 3 ). This shows that ( r = 3 ) is the factor by which each term is multiplied to obtain the next term.

Recognizing this property of ( r ) is crucial for understanding how geometric sequences function, enabling calculations of specific terms and the overall behavior of the sequence. It's also important to differentiate ( r ) from other concepts, such as the common difference found in arithmetic sequences, where each term is derived from adding or subtracting a fixed number rather than multiplying by a ratio. The first term and last term are simply positional aspects of the sequence, not representative of the pattern that defines