In pre-calculus, sequences and series are important concepts that deal with ordered lists of numbers and the sum of those numbers.

Key Points:

• Sequence: A sequence is an ordered list of numbers. The numbers in a sequence are called terms. Sequences can be finite (with a specific number of terms) or infinite (with an endless number of terms).
• Series: A series is the sum of the terms in a sequence. If you add up all the terms in a sequence, you get a series.

Types of Sequences:

• Arithmetic Sequence: In an arithmetic sequence, the difference between consecutive terms is constant. This difference is called the common difference.
• Geometric Sequence: In a geometric sequence, each term is found by multiplying the previous term by a constant. This constant is called the common ratio.

Example 1: Arithmetic Sequence

Consider the sequence: 2, 5, 8, 11, 14, .... This is an arithmetic sequence because the difference between each term is the same. The common difference is 3.
To find the next term, we add 3 to the last term: 14 + 3 = 17.
This sequence continues infinitely by adding 3 to each term.

Example 2: Geometric Sequence

Consider the sequence: 3, 6, 12, 24, 48, .... This is a geometric sequence because each term is obtained by multiplying the previous term by 2, which is the common ratio.
To find the next term, we multiply the last term by 2: 48 × 2 = 96.
This sequence continues infinitely by multiplying each term by 2.

Series:

A series is the sum of the terms in a sequence. For example:
For the arithmetic sequence 2, 5, 8, 11, the sum of the first four terms is:
2 + 5 + 8 + 11 = 26.
For the geometric sequence 3, 6, 12, the sum of the first three terms is:
3 + 6 + 12 = 21.

Formula for the Sum of an Arithmetic Series:

The sum of the first n terms of an arithmetic sequence is given by the formula:
Sₙ = n/2 × (2a + (n - 1) × d)
where Sₙ is the sum, a is the first term, n is the number of terms, and d is the common difference.

Example 3: Arithmetic Series

Using the arithmetic sequence 2, 5, 8, 11, let's find the sum of the first 4 terms:
S₄ = 4/2 × (2(2) + (4 - 1) × 3) = 2 × (4 + 9) = 2 × 13 = 26.
So, the sum of the first 4 terms is 26.

Summary:

A sequence is an ordered list of numbers, and a series is the sum of the terms in a sequence. Sequences can be arithmetic, where each term is found by adding a constant difference, or geometric, where each term is found by multiplying by a constant ratio. Understanding sequences and series helps to model real-world situations and solve problems involving patterns of numbers.

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