What is the difference between a sequence and a series?
We can consider sequences and series as superheroes that solve puzzles! They assist us in deciphering codes and resolving mysteries, much like detectives. Sequences function as preliminary hints, whereas series represent the main players providing major solutions. We can imagine them as friends solving math puzzles together to reveal the solutions, adding excitement and enjoyment to solving problems.
Sequence
A sequence is an ordered list of numbers. A term is any one of the sequence’s numbers. Sequence terms are arranged in a particular order and might be infinite or finite.
The general form of a sequence is denoted as {a₁, a₂, a₃, ...}, where “a₁” and “a₂” stand for the first and second terms. The numerals 1, 2, and 3 denote the terms’ positions.
Example
The
sequence is an arithmetic series with a common difference of two. The
sequence is a geometric sequence with a common ratio of two.
To illustrate, a car’s resale value that drops by $500 annually, and an employee’s annual income that increases by $2,000 follows an arithmetic sequence. On the other hand, geometric growth can be observed in the case of a town’s population tripling every ten years, while geometric investment development is represented by a savings account with a 5% interest rate.
Series
A series is the sum of a sequence’s terms, which is created by totaling the terms. A series is equally finite if the sequence is finite too. In the case where the sequence is infinite, the series might diverge (not approach any particular value) or converge to a certain value.
The “a₁ + a₂ + a₃ +...” form which represents the total of all the items in the sequence, is the general form of a series. The terms in a series are represented by the numbers 1, 2, 3, and so on.
Examples
The arithmetic series for the mathematical sequence
is (the sum of the terms). The series that corresponds to a geometric sequence
is (the total of the terms).
An arithmetic series can be formed by the cumulative earnings with a $2,000 annual wage increment and a $500 annual decrease in car value. An example of a geometric series can include the town’s population tripling every ten years and the growth of savings account balancing with a 5% interest rate.
Graphical representation
Arithmetic sequences are represented graphically as a straight line highlighting the constant difference between terms that come after one another. A linear cumulative sum curve, representing constant linear growth, is shown in the associated arithmetic series graph. On the other hand, a geometric series is represented by an exponential cumulative total curve that emphasizes the compounding impact and exponential increase over subsequent terms, while geometric sequences display an exponential pattern with varied distances between points.
Conclusion
To summarize, a series is the total of all the terms in a sequence, whereas a sequence is an ordered list of numbers. Sequences deal with how individual pieces are arranged, whereas series combine these elements by adding new ones.
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