The general rule of a geometric sequence allows you to find any term in the sequence without having to calculate all the preceding terms. It's expressed in the form an = a1 * r(n-1), where an is the nth term, a1 is the first term, r is the common ratio, and n is the term number.
Here's a breakdown of how to find the general rule:
1. Identify the First Term (a1):
- The first term is simply the first number in the sequence.
2. Determine the Common Ratio (r):
- The common ratio is the constant value you multiply each term by to get the next term. You can find it by dividing any term by its preceding term: r = a2/a1 = a3/a2, and so on.
3. Write the General Rule:
- Substitute the values of a1 and r into the formula: an = a1 * r(n-1). This is your general rule.
Example:
Consider the geometric sequence: 2, 6, 18, 54, ...
- a1 (First term) = 2
- r (Common Ratio) = 6/2 = 18/6 = 3
Therefore, the general rule for this geometric sequence is: *an = 2 3(n-1)**
Finding the Rule When Given Two Terms:
Sometimes, you might be given two terms of the sequence but not the first term. Here's how to find the general rule in that case, based on the transcribed video snippet:
1. Use the General Formula for Both Terms:
Let's say you're given that the 3rd term (a3) is 4 and the first term (a1) is 9. We know an = a1 * r(n-1)
- a3 = a1 r(3-1) => 4 = 9 r2
2. Solve for the Common Ratio (r):
- Divide both sides by 9: 4/9 = r2
- Take the square root of both sides: r = ±√(4/9) = ±2/3
3. Write the General Rule(s):
- Since we have two possible values for r, we have two possible general rules:
- an = 9 * (2/3)(n-1)
- an = 9 * (-2/3)(n-1)
Important Considerations:
- Geometric sequences can have a positive or negative common ratio. A negative common ratio means the terms alternate in sign.
- Be careful with the order of operations. Remember to calculate the exponent (r(n-1)) before multiplying by a1.
