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How to Find the Common Ratio of a Geometric Sequence with Fractions?

Published in Geometric Sequences 3 mins read

To find the common ratio of a geometric sequence, even when dealing with fractions, you can divide any term by the term that precedes it.

Here’s how to approach this:

  • Understanding the Concept: In a geometric sequence, each term is found by multiplying the previous term by a constant value, which is called the common ratio. This constant multiplier doesn't change throughout the sequence.

  • The Formula: If you have a geometric sequence with terms a1, a2, a3, ..., the common ratio (r) can be found by:

    • r = a2 / a1
    • r = a3 / a2
    • And so on...

  • Dealing with Fractions: The same principle applies when you have fractions in your sequence. You still divide any term by the term that came before it.

Steps to Find the Common Ratio

  1. Identify two consecutive terms: Choose any two terms that are next to each other in the sequence. Let's say we have an and an-1.

  2. Divide the later term by the earlier term: Calculate an / an-1. This operation will give you the common ratio (r).

  3. Verify: To ensure you found the correct ratio, you should perform the division on another set of consecutive terms.

Example

Let's consider a geometric sequence with fractions: 1/2, 1/4, 1/8, 1/16, ...

  1. Identify consecutive terms: Let’s pick 1/4 and 1/2.

  2. Divide: (1/4) / (1/2)

    • Remember that dividing by a fraction is the same as multiplying by its reciprocal: (1/4) * (2/1)
    • (1 2) / (4 1) = 2 / 4
    • Simplify: 2/4 = 1/2

  3. Verify: Now let's check with another pair, 1/8 and 1/4:

    • (1/8) / (1/4) = (1/8) * (4/1)
    • (1 4) / (8 1) = 4 / 8
    • Simplify: 4/8 = 1/2
    • The ratio is consistent.

Therefore, in this sequence, the common ratio is 1/2.

Reference Insight

As mentioned in the provided reference [Part of a video titled Learning to find the ratio of a geometric sequence - YouTube], it might be tricky to divide by fractions, so it can be helpful to think of it as multiplying by the reciprocal of the fraction. This tip can simplify the calculation process, making it easier to determine the common ratio when dealing with fractions.

In summary, to find the common ratio of a geometric sequence involving fractions, simply divide any term by its preceding term. Converting division by a fraction to multiplication by its reciprocal can be a useful technique.

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