Series
5 Problems • 2 sub-topics
Adalynn Le • 5/25/2026
Introduction
Series are the sum of sequences. On their own, sequences are not all that complicated and can be modeled with equations. Sequences model their sums and can even bound infinite sequences with specific constraints.
Arithmetic Series
Arithmetic Sequences
Arithmetic sequences grow linearly. They have a principle value \(a\) and a common difference \(d\), such that \(a_{n}-a_{n-1}=d\). The value of \(a_{n}=a_{1}+ n \times d\). This is similar to the linear function \(y=mx+b\), with the input \(x\) effectively the same as \(n\). Any linear relationship is an arithmetic series.
Sum
The sum of an arithmetic series, denoted as \(\sum_{i=1}^{n} a_{n}\) can be solved in different ways depending on if we know \(a_{n}\) or not. Notice that arithmetic sequences always grow, or get progressively smaller, they are not bounded. Thus, we cannot find an infinite sum and must constrain the sum. If we do know \(a_{n}\), we have \(n(\frac{a_{1}+a_{n}}{2}\)). Notice that this is very similar to the sum of all integers \(leq n\) which is \(\frac{n(n+1)}{2}\), because it is. All integers in order make an arithmetic sequence, and the sum of this is has \(a_{1}=1\) and \(a_{n}=n\)
Geometric Series
Geometric Sequences
Just as a geometric sequence models a linear relation with a common difference, a geometric series models an exponential relationship with a common ratio. The value \(a_{n}\) of a geometric sequence can be modeled as \(a_{n}=a \times r^n\). Just as the arithmetic sequence is graphed as a linear relationship, a geometric sequence can be measured with a exponential function. A common variant on geometric sequences-unlikely to come up on the AMC 10 but good to know-is compound interest, the growth of money invested by interest. This can be modeled as \(a(1 +\frac{r}{n})^{nt}\)
Sum
The sum of a geometric series depends on whether it is said to be "convergent" or "divergent". Convergent series have \(|r| \leq 1\), meaning that oer time, they will begin to get closer and closer to \(0\). The sum for this would be \(\frac{a_{1}}{1-r}\). As you can see, there is no value for \(n\), because it should not be dependent on \(n\) because it is innfinite. A divergent, or finite series, is bounded by \(n\) and has \(|r| \geq 1\). The sum for this is \(a_{1}(\frac{1-r^n}{1-r})\)
Conclusion
Series and sequences can model growth in different ways. The rate or difference is what determines the type of sequence and how it is modeled. Arithmetic series must be bound by \(n\), but geometric series might not be if they are infinite and converge at a certain value.
Question 1:
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