Math Analysis & Approaches
Numbers & Algebra
Sequences & Series
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Sequences & Series
Introduction to Sequences
- A sequence is a list of numbers in a specific order, where each number is called a term of the sequence.
- Sequences can be finite (having a definite number of terms) or infinite (never ending).
- The notation a<sub>n</sub> often represents the nth term of a sequence.
- The difference between consecutive terms characterises arithmetic sequences. This difference is constant and is called the common difference.
- In geometric sequences, each term is a fixed multiple of the previous term. This multiple is constant and is called the common ratio.
- Formulas for the nth term exist for both arithmetic and geometric sequences: a<sub>n</sub> = a + (n-1)d for arithmetic sequences and a<sub>n</sub> = ar<sup>n-1</sup> for geometric sequences, where a is the first term, d is the common difference, r is the common ratio, and n is the term number.
Understanding Series
- A series is the sum of the terms in a sequence.
- The sum of the first n terms in a series is often represented as S<sub>n</sub>.
- There are formulas to calculate the sum of the first n terms in arithmetic and geometric sequences: S<sub>n</sub> = n/2 (2a + (n-1)d) for arithmetic series and S<sub>n</sub> = a (1-r<sup>n</sup>)/(1-r) for geometric series when |r|<1.
- In the case of geometric series where |r|≥1, the sum of the terms approaches infinity as n increases; this is called a divergent series. When |r|<1, the sum of the terms approaches a finite limit as n increases; this is called a convergent series.
Working with Sequences and Series
- Recursive sequences are those wherein each term is defined based on previous terms. The Fibonacci sequence is a classical example of this.
- Sequences and series can be found in various fields and concepts such as finance (calculating compound interest), physics (calculating distance travelled under constant acceleration), and computer algorithms (generating number sequences or patterns).
- Real-life applications often involve infinite geometric series, with their sums representing a physical total or limit (for example, the total distance covered in a constantly decelerating process).
- You can graph the terms of a sequence to demonstrate the relationship between the terms. Arithmetic sequences yield a linear graph, while geometric sequences yield an exponential graph with either growth or decay depending on the common ratio.
Finding Limits
- The limit of a sequence (L) is the value the sequence gets closer to as the number of terms increases. In other words, if a sequence gets closer and closer to a particular value as n becomes very large, then that value is the limit of the sequence.
- The limit of an arithmetic sequence is either infinity, negative infinity, or non-existent because the sequence just keeps growing or decreasing.
- The limit of a geometric sequence is dependent on the common ratio, r. If |r|<1, then the limit is 0; if r=1, the limit is the first term, a; if r>1 or r<-1, the limit does not exist (sequence diverges). For example, the sequence defined by a<sub>n</sub> = (1/2)<sup>n-1</sup> has a limit of 0.
- Convergence or divergence of sequences are important concepts when dealing with infinite series. An infinite series will only have a finite sum (i.e., converge) if the sequence it is based on has a limit of zero.
