Arithmetic sequences are number patterns where each term differs from the previous by a constant amount. This constant difference is the key to understanding and predicting these sequences, whether they're increasing, decreasing, or staying the same.

You can describe arithmetic sequences using recursive or explicit formulas. Recursive formulas define each term based on the previous one, while explicit formulas let you find any term directly. These formulas are super useful for solving real-world problems involving linear growth or patterns.

Arithmetic Sequences

Common difference in arithmetic sequences

• Constant value added to each term to obtain the next term in the sequence
• Calculated by subtracting any term from the subsequent term $d = a_{n+1} - a_n$
• Can be positive (increasing sequence), negative (decreasing sequence), or zero (constant sequence)
• Example: In the sequence (progression) 2, 5, 8, 11, 14, the common difference is 3 (5 - 2 = 8 - 5 = 11 - 8 = 14 - 11 = 3)

Recursive formulas for arithmetic sequences

• Define each term based on the previous term and the common difference
• General form: $a_{n+1} = a_n + d$, where $a_{n+1}$ is the next term, $a_n$ is the current term, and $d$ is the common difference
• Requires knowing the first term ($a_1$) and the common difference ($d$)
• Example: For the sequence 3, 7, 11, 15, 19, the recursive formula is $a_{n+1} = a_n + 4$, with $a_1 = 3$

Explicit formulas for arithmetic sequences

• Express the $n$th term directly in terms of $n$, the first term ($a_1$), and the common difference ($d$)
• General form: $a_n = a_1 + (n - 1)d$, where $a_n$ is the $n$th term, $a_1$ is the first term, $n$ is the term's position, and $d$ is the common difference
• Allows finding any term without calculating the previous terms
• Example: For the sequence 5, 9, 13, 17, 21, the explicit formula is $a_n = 5 + (n - 1)4$, where $a_1 = 5$ and $d = 4$
• The explicit formula represents a linear relationship between the term number and its value

Real-world applications of arithmetic sequences

• Modeling linear growth or decline (population growth, depreciation)
• Financial planning (regular savings, investments, or payments)
• Seating arrangements (auditoriums, stadiums)
• Example: A concert hall has 20 seats in the first row and each subsequent row has 2 more seats than the previous row. The seating arrangement follows an arithmetic sequence with $a_1 = 20$ and $d = 2$

Solving problems with arithmetic sequences

1. Identify the initial value and common difference
2. Choose the appropriate formula (recursive or explicit) based on the problem
3. Substitute the given values into the formula
4. Solve for the unknown variable and interpret the result

Additional concepts in arithmetic sequences

• Sequence: An ordered list of numbers following a specific pattern
• Term: Each individual number in a sequence
• Summation: The process of adding up all terms in a sequence
• Arithmetic mean: The average of two terms in an arithmetic sequence, which is always equal to the term halfway between them