To find the common ratio (r) of an infinite geometric progression, you typically need to use the sum to infinity formula or examine consecutive terms in the sequence.
Finding the Common Ratio
Here are two primary methods:
1. Using the Sum to Infinity Formula
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The Formula: The sum to infinity (S) of a geometric progression is given by: S = a / (1 - r), where 'a' is the first term and 'r' is the common ratio. This formula is valid only if |r| < 1.
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Rearranging the Formula: If you know the sum to infinity (S) and the first term (a), you can rearrange the formula to solve for 'r':
r = 1 - (a / S)
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Example: Suppose the sum to infinity of a geometric progression is 9, and the first term is 1. Then, r = 1 - (1/9) = 8/9. As illustrated by the reference video, this involves setting up the equation 1 / (1 - r) = 9 and solving for r.
2. Examining Consecutive Terms
- The Method: In any geometric progression, the common ratio 'r' is the constant value you multiply one term by to get the next term.
- Calculation: Choose any two consecutive terms, say the n-th term (an) and the (n+1)-th term (an+1). Then, the common ratio r = an+1 / an.
- Example: If the first few terms of a geometric progression are 2, 1, 0.5, 0.25,..., then you can find the common ratio by dividing any term by its preceding term. For example, r = 1/2 = 0.5, or r = 0.5/1 = 0.5, or r = 0.25/0.5 = 0.5.
Important Notes:
- For the sum to infinity formula to be applicable, the absolute value of the common ratio must be less than 1 (|r| < 1). If |r| ≥ 1, the series diverges, and the sum to infinity does not exist.
- When examining consecutive terms, make sure to check several pairs to confirm that the ratio is consistent throughout the sequence. This ensures it is indeed a geometric progression.
