3.10 Arithmetic Sequences

Arithmetic sequences are number lists with a constant difference between terms. They're easy to spot and have a handy formula for finding any term. This makes them super useful for predicting patterns and solving real-world problems.

Knowing how to calculate the sum of a finite arithmetic sequence is a game-changer. It lets you quickly add up large sets of numbers, which comes in handy for everything from finance to physics calculations.

Arithmetic Sequences

Recognition of arithmetic sequences

• Arithmetic sequence is a list of numbers where the difference between any two consecutive terms remains constant
• This constant difference called the common difference denoted as $d$
• To identify an arithmetic sequence, calculate the difference between each pair of consecutive terms
  • If the differences are equal for all pairs, the sequence is arithmetic (1, 4, 7, 10, ...) since the common difference is 3
  • If the differences are not equal, the sequence is not arithmetic (1, 4, 9, 16, ...) since the differences are not constant

Formula for nth term

• The formula for the nth term of an arithmetic sequence is $a_n = a_1 + (n - 1)d$
  • $a_n$ represents the nth term in the sequence
  • $a_1$ represents the first term in the sequence
  • $n$ represents the position of the term
  • $d$ represents the common difference between consecutive terms
• To find a specific term in an arithmetic sequence using the formula:
  1. Identify the first term $a_1$, the common difference $d$, and the position $n$ of the desired term
  2. Substitute these values into the formula $a_n = a_1 + (n - 1)d$ and simplify
• Example: Find the 8th term of the arithmetic sequence (also known as an arithmetic progression) 5, 9, 13, 17, ...
  • $a_1 = 5$, $d = 4$ (since 9 - 5 = 4, 13 - 9 = 4, etc.), and $n = 8$
  • $a_8 = 5 + (8 - 1)4 = 5 + 28 = 33$
  • The 8th term of the sequence is 33

Sum of finite arithmetic sequences

• The sum of a finite arithmetic sequence can be calculated using the formula $S_n = \frac{n}{2}(a_1 + a_n)$
  • $S_n$ represents the sum of the first $n$ terms in the sequence
  • $a_1$ represents the first term in the sequence
  • $a_n$ represents the nth term in the sequence
  • $n$ represents the number of terms being added
• To find the sum of a finite arithmetic sequence using the formula:
  1. Identify the first term $a_1$, the last term $a_n$, and the number of terms $n$
  2. Substitute these values into the formula $S_n = \frac{n}{2}(a_1 + a_n)$ and simplify
• Example: Find the sum of the first 12 terms of the arithmetic sequence 3, 7, 11, 15, ...
  • $a_1 = 3$, $n = 12$, and the common difference $d = 4$
  • To find $a_{12}$, use the formula $a_n = a_1 + (n - 1)d$
    • $a_{12} = 3 + (12 - 1)4 = 3 + 44 = 47$
  • Substitute the values into the sum formula: $S_{12} = \frac{12}{2}(3 + 47) = 6(50) = 300$
  • The sum of the first 12 terms is 300

Additional Concepts

• Sequence: An ordered list of numbers following a specific pattern
• Series: The sum of the 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