Arithmetic sequences are like climbing stairs, where each step is the same height. The is that constant step, and you can use it to find any in the sequence. It's a simple yet powerful concept in math.
Understanding arithmetic sequences unlocks the ability to predict patterns and solve real-world problems. Whether you're calculating interest or analyzing data trends, these sequences provide a foundation for more complex mathematical concepts.
Arithmetic Sequences
Common difference calculation
Calculate common difference d by subtracting any term from the next term in the sequence an+1−an=d
Alternatively, use formula d=n−1an−a1 given first term a1, last term an, and total number of terms n
Example: In sequence 2,5,8,11,..., d=5−2=3 or d=4−111−2=3
Term generation in arithmetic sequences
Generate next term by adding common difference d to previous term an+1=an+d
Sequence defined by first term a1 and common difference d
An is a type of sequence where each term differs from the previous by a constant amount
Also known as an arithmetic progression, it follows a consistent pattern of addition or subtraction
Each term in the sequence can be generated using the term position and common difference
The sequence forms a linear pattern when graphed, with each term increasing or decreasing at a constant rate
Key Terms to Review (4)
Arithmetic sequence: An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This fixed difference is called the common difference.
Common difference: The common difference is the constant amount that each term in an arithmetic sequence differs from the previous term. It is denoted by 'd' and can be positive, negative, or zero.
Recursive formula: A recursive formula defines each term of a sequence using the preceding term(s). It typically includes an initial condition and a recurrence relation.
Term: An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This difference is called the common difference.