An arithmetic sequence is like climbing stairs where each step is the same height. You add the same amount each time to get the next number. This fixed amount is called the common difference. For example, in the sequence 2, 4, 6, 8, 10, we add 2 each time.
A geometric sequence is like something doubling or tripling repeatedly. You multiply by the same amount each time to get the next number. This fixed amount is called the common ratio. For example, in the sequence 3, 6, 12, 24, 48, we multiply by 2 each time.
The key difference between these sequences is simple. Arithmetic sequences use addition of a constant value, creating linear growth like a straight line. Geometric sequences use multiplication by a constant value, creating exponential growth like a curve that gets steeper and steeper.
Let's look at more examples. For arithmetic sequences, we can subtract instead of add, like 10, 7, 4, 1, negative 2, where we subtract 3 each time. For geometric sequences, we can multiply by fractions or negative numbers, like 2, 6, 18, 54, where we multiply by 3 each time.
To summarize, remember this simple rule: Arithmetic sequences use addition with the same difference each step, like climbing stairs of equal height. Geometric sequences use multiplication with the same ratio each step, like something doubling or halving. Both are just different patterns of growth that we see everywhere in mathematics and real life.