Integrated Math 1: Arithmetic Sequences – The Ultimate Step-by-Step Guide

🤔 What are Arithmetic Sequences?

An arithmetic sequence is a list of numbers where the difference between any two consecutive terms is constant. This constant difference is called the common difference, often denoted as 'd'.

➕ The Formula

The general formula for the nth term ($a_n$) of an arithmetic sequence is:

$a_n = a_1 + (n - 1)d$

• $a_n$ = the nth term
• $a_1$ = the first term
• n = the term number
• d = the common difference

✍️ Finding the Common Difference (d)

To find the common difference, subtract any term from its subsequent term:

$d = a_2 - a_1 = a_3 - a_2 = ...$

✔️ Examples

Example 1: Finding the nth term

Consider the arithmetic sequence: 2, 5, 8, 11, ... Find the 10th term.

1. Identify $a_1$ and d: $a_1 = 2$, $d = 5 - 2 = 3$
2. Apply the formula: $a_{10} = 2 + (10 - 1) * 3$
3. Calculate: $a_{10} = 2 + 9 * 3 = 2 + 27 = 29$

Therefore, the 10th term is 29.

Example 2: Finding a Missing Term

In an arithmetic sequence, the 3rd term is 7 and the 7th term is 15. Find the first term and the common difference.

1. Set up equations:
   • $a_3 = a_1 + 2d = 7$
   • $a_7 = a_1 + 6d = 15$
2. Solve the system of equations: Subtract the first equation from the second:

   $(a_1 + 6d) - (a_1 + 2d) = 15 - 7$

   $4d = 8$

   $d = 2$

3. Substitute d back into the first equation:

   $a_1 + 2(2) = 7$

   $a_1 + 4 = 7$

   $a_1 = 3$

Therefore, the first term is 3 and the common difference is 2.

💻 Code Example (Python)

Here's a Python function to find the nth term of an arithmetic sequence:

def nth_term_arithmetic_sequence(a1, d, n):
  """Calculates the nth term of an arithmetic sequence.

  Args:
    a1: The first term of the sequence.
    d: The common difference.
    n: The term number to find.

  Returns:
    The nth term of the sequence.
  """
  an = a1 + (n - 1) * d
  return an

# Example usage:
first_term = 2
common_difference = 3
term_number = 10
nth_term = nth_term_arithmetic_sequence(first_term, common_difference, term_number)
print(f"The {term_number}th term is: {nth_term}") # Output: The 10th term is: 29

💪 Practice Problems

1. Find the 15th term of the arithmetic sequence: 1, 4, 7, 10, ...
2. The 5th term of an arithmetic sequence is 22, and the common difference is 3. Find the first term.
3. Find the common difference of an arithmetic sequence where the 2nd term is 8 and the 9th term is 29.

🔑 Solutions to Practice Problems

1. $a_{15} = 1 + (15-1) * 3 = 1 + 14 * 3 = 1 + 42 = 43$
2. $a_5 = a_1 + 4d \Rightarrow 22 = a_1 + 4 * 3 \Rightarrow 22 = a_1 + 12 \Rightarrow a_1 = 10$
3. $a_9 = a_2 + 7d \Rightarrow 29 = 8 + 7d \Rightarrow 21 = 7d \Rightarrow d = 3$