Finding the nth term of a geometric sequence is straightforward using a simple formula. A geometric sequence is a list of numbers where each term after the first is found by multiplying the previous one by a constant, called the common ratio.
The Formula
The formula to find the nth term (an) of a geometric sequence is:
*an = a1 r(n-1)**
Where:
- an is the nth term you want to find.
- a1 is the first term of the sequence.
- r is the common ratio (the number you multiply each term by to get the next).
- n is the term number you're looking for.
Understanding the Components
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a1: This is simply the first number in your geometric sequence. For example, in the sequence 2, 6, 18, 54..., a1 = 2.
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r: To find the common ratio, divide any term by the term before it. In the example sequence above, 6/2 = 3, 18/6 = 3, and 54/18 = 3. Therefore, r = 3.
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n: This is the position of the term you wish to calculate within the sequence. If you want the 5th term, n = 5.
Examples
Let's illustrate with a few examples:
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Example 1: Find the 5th term (a5) of the geometric sequence 3, 6, 12, 24...
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a1 = 3
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r = 6/3 = 2
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n = 5
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a5 = 3 2(5-1) = 3 24 = 3 * 16 = 48
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Example 2: Find the 8th term (a8) of the geometric sequence 1, 1/2, 1/4, 1/8...
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a1 = 1
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r = (1/2) / 1 = 1/2
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n = 8
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a8 = 1 * (1/2)(8-1) = (1/2)7 = 1/128
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Resources for Further Learning
Numerous online resources are available to help you master geometric sequences. You can explore videos on YouTube such as "How to Find the nth Term of a Geometric Sequence" or utilize online calculators designed to find the nth term. Remember to always clearly identify a1 and r before applying the formula.
