Geometric sequences

• What is a geometric sequence?


  A geometric sequence is a list of numbers where each term is multiplied by a common ratio.
• What is the difference between a sequence and a series?


  A sequence is a list of numbers, while a series is the sum of the terms of a sequence.
• How is an arithmetic sequence characterized?


  An arithmetic sequence is characterized by having a common difference between all its terms.
• What is the pattern in the geometric sequence 2, 4, 8, 16?


  The pattern is that each term is multiplied by 2 to get the next term.
• What do we call the common factor in a geometric sequence?


  The common factor in a geometric sequence is called the common ratio, denoted as \( r \).
• How can the common ratio \( r \) be found in a geometric sequence?


  The common ratio \( r \) can be found by dividing any term by the term before it.
• What is the formula for the nth term of a geometric sequence?


  The formula for the nth term is \( u_n = u_1 \cdot r^{n-1} \).
• How would you find the 10th term of a geometric sequence with first term \( u_1 = -2 \) and common ratio \( r = -3 \)?


  You would calculate \( u_{10} = -2 \cdot (-3)^{10-1} \).
• Why is it important to follow the order of operations in calculations?


  It is important to follow the order of operations to ensure accurate results in mathematical calculations.
• What does the acronym PEMDAS stand for in the order of operations?


  PEMDAS stands for Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).
• How would you calculate \( u_{10} \) using a calculator for the sequence with \( u_1 = -2 \) and \( r = -3 \)?


  You would input \( -2 \cdot (-3)^9 \) into the calculator.
• In what real-world applications are geometric sequences used?

  • Nuclear physics
  • Finance for loans (compound interest)
  • Radioactive decay
• How does the calculation of the 10th term in a geometric sequence differ from that in an arithmetic sequence?


  In a geometric sequence, you multiply by the common ratio, while in an arithmetic sequence, you add the common difference.