A sequence is an ordered list of numbers, where each number is called a term.
Sequences can be finite(having a specific number of terms) or infinite(continuing indefinitely).
Explicit Formula for a Sequence
An explicit formula defines the \$n\$-th term of a sequence directly in terms of \$n\$.
The explicit formula allows you to find any term in the sequence without knowing the previous terms.
Recursive formulas require a base caseto start the sequence.
Recursive formulas can be less efficient for finding terms far into the sequence, as they require calculating all previous terms.
Arithmetic and Geometric Sequences
Arithmetic Sequences
A sequence in which the difference between consecutive terms is constant.
The constant difference is called the common differenceand is denoted by \$d\$.
The constant ratio is called the common ratioand is denoted by \$r\$.
Series can be finite(summing a specific number of terms) or infinite(summing indefinitely).
Finite Arithmetic Series
The sum of the first \$n\$ terms of an arithmetic sequence is given by:
\$\$S_n = \frac{n}{2} (a_1 + a_n)\$\$
1. Find the 15th term of the arithmetic sequence \$3, 7, 11, 15, \ldots\$. 2. Find the 8th term of the geometric sequence \$2, 6, 18, 54, \ldots\$. 3. Calculate the sum of the first 20 terms of the arithmetic sequence \$4, 9, 14, 19, \ldots\$. 4. Calculate the sum of the first 5 terms of the geometric sequence \$1, 3, 9, 27, \ldots\$.
How do sequences and series help us model real-world situations? Can you think of examples where these concepts are applied outside of mathematics?
Be careful when using recursive formulas. Always ensure you have the correct base case and understand how the terms are related.
When working with sequences and series, always identify whether the sequence is arithmetic or geometric. This will guide you in choosing the correct formula.
