Geometric sequences

Learning Objectives Define a geometric sequence and identify its key components: the first term (a₁) and the common ratio (r). Calculate the common ratio of a geometric sequence. Write the explicit formula for the nth term of a geometric sequence. Use the nth term formula to find any specific term in a geometric sequence. Distinguish between a geometric sequence and an arithmetic sequence. Model a real-world situation, such as compound interest or population decay, using a geometric sequence. Ever wonder how a video goes viral, doubling its views every hour? That's a geometric sequence in action! 📈 In this lesson, you'll discover geometric sequences, where you multiply by the same number over and over to get the next term. This powerful idea helps us understand a...

TermDefinitionExample SequenceAn ordered list of numbers, called terms, that follow a specific pattern or rule.2, 4, 6, 8, ... is a sequence where you add 2 each time. Geometric SequenceA sequence where each term after the first is found by multiplying the previous term by a fixed, non-zero number.3, 9, 27, 81, ... is a geometric sequence because you multiply by 3 each time. First Term (a₁)The very first number in a sequence.In the sequence 5, 10, 20, 40, ..., the first term (a₁) is 5. Common Ratio (r)The fixed number that you multiply by to get from one term to the next in a geometric sequence.In the sequence 100, 50, 25, 12.5, ..., the common ratio (r) is 0.5 (or 1/2). nth Term (aₙ)A variable that represents any term in the sequence, based on its position 'n'.In the sequence 2...

Finding the Common Ratio r = aₙ / aₙ₋₁ To find the common ratio (r), divide any term by the term that comes directly before it. For example, you can divide the second term by the first term (a₂ / a₁) or the third term by the second term (a₃ / a₂). The nth Term Formula for a Geometric Sequence aₙ = a₁ * r^(n-1) This is the explicit formula to find any term (aₙ) in a geometric sequence. You need to know the first term (a₁), the common ratio (r), and the position of the term you want to find (n).