12.2 Arithmetic Sequences

Arithmetic sequences are number patterns with a constant difference between terms. They're like stepping stones, each one a fixed distance from the last. This concept is key to understanding patterns in math and real-world situations.

Learning about arithmetic sequences helps you predict future terms and calculate sums efficiently. It's a building block for more complex mathematical ideas and has practical applications in finance, physics, and everyday problem-solving.

Arithmetic Sequences

Patterns in arithmetic sequences

• Arithmetic sequence (also known as arithmetic progression) defined as sequence of numbers with constant difference between consecutive terms
• Constant difference called common difference denoted as $d$
• Identify arithmetic sequence by examining if each term obtained by adding fixed value to previous term
  • Sequence 2, 5, 8, 11, 14, ... is arithmetic with common difference of 3 since each term is 3 more than previous term
• Other examples of arithmetic sequences:
  • 1, 4, 7, 10, 13, ... (common difference of 3)
  • 10, 7, 4, 1, -2, ... (common difference of -3)

General term formula construction

• General term formula for arithmetic sequence is $a_n = a_1 + (n - 1)d$
  • $a_n$ represents the $n$th term of sequence
  • $a_1$ represents the first term of sequence
  • $n$ represents the position of term in sequence
  • $d$ represents the common difference
• Construct general term formula by identifying first term ($a_1$) and common difference ($d$)
  • Substitute values into formula $a_n = a_1 + (n - 1)d$
• Example: Sequence 2, 5, 8, 11, 14, ... has general term formula $a_n = 2 + (n - 1)3$ or simplified as $a_n = 3n - 1$
  • First term ($a_1$) is 2 and common difference ($d$) is 3
• Another example: Sequence 10, 7, 4, 1, -2, ... has general term formula $a_n = 10 + (n - 1)(-3)$ or simplified as $a_n = 13 - 3n$
  • First term ($a_1$) is 10 and common difference ($d$) is -3

Sum of finite arithmetic sequences

• Sum of finite arithmetic sequence calculated using formula $S_n = \frac{n}{2}(a_1 + a_n)$
  • $S_n$ represents sum of first $n$ terms of sequence
  • $a_1$ represents first term of sequence
  • $a_n$ represents $n$th term of sequence
  • $n$ represents number of terms in sequence
• Calculate sum by identifying first term ($a_1$), last term ($a_n$), and number of terms ($n$)
  • Substitute values into formula $S_n = \frac{n}{2}(a_1 + a_n)$
• Alternative method to calculate sum:
  1. Find general term formula $a_n = a_1 + (n - 1)d$
  2. Substitute value of $a_n$ into sum formula $S_n = \frac{n}{2}(a_1 + a_n)$
• Example: Find sum of first 10 terms of sequence 2, 5, 8, 11, 14, ...
  • $a_1 = 2$, $a_{10} = 29$ (calculated using general term formula), and $n = 10$
  • $S_{10} = \frac{10}{2}(2 + 29) = 5(31) = 155$
• Another example: Find sum of first 6 terms of sequence 10, 7, 4, 1, -2, ...
  • $a_1 = 10$, $a_6 = -8$ (calculated using general term formula), and $n = 6$
  • $S_6 = \frac{6}{2}(10 + (-8)) = 3(2) = 6$

Types of Sequences and Series

• A sequence is an ordered list of numbers
• Finite sequences have a definite end, while infinite sequences continue indefinitely
• A series is the sum of the terms in a sequence
• Arithmetic sequences can be either finite or infinite, depending on the context