A sequence is an ordered list of numbers, while a partial sum is the sum of a finite number of consecutive terms from that sequence.
Here's a more detailed explanation:
Understanding Sequences
A sequence is simply a list of elements (usually numbers) arranged in a specific order. Each element in the sequence is called a term. A sequence can be finite (ending after a certain number of terms) or infinite (continuing indefinitely).
Example:
The sequence 2, 4, 6, 8, 10, ... is an infinite sequence of even numbers.
Understanding Partial Sums
A partial sum, on the other hand, is the sum of a certain number of terms from a sequence, starting from the first term. Because we are only summing part of a sequence, it's called a partial sum.
Example (using the sequence above):
- The first partial sum is just the first term:
2 - The second partial sum is the sum of the first two terms:
2 + 4 = 6 - The third partial sum is the sum of the first three terms:
2 + 4 + 6 = 12 - And so on...
Key Differences Summarized
| Feature | Sequence | Partial Sum |
|---|---|---|
| Definition | An ordered list of numbers. | The sum of a finite number of terms from a sequence. |
| Result | A list of numbers. | A single number (the sum). |
| Dependency | Independent; defines the list itself. | Dependent on a pre-existing sequence. |
In essence, a sequence provides the numbers, and the partial sum operates on those numbers to produce a single numerical value. A sequence is a list, while a partial sum is the sum of a part of that list.
