To find the sum of the first 10 terms in a geometric sequence, you can use a specific formula that accounts for the common ratio between terms.
Understanding Geometric Sequences
A geometric sequence is a series of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio (r). For example, 2, 6, 18, 54,... is a geometric sequence with a common ratio of 3.
The Formula for the Sum of a Geometric Series
The sum (Sₙ) of the first n terms of a geometric series can be calculated using the formula:
Sₙ = a * (1 - rⁿ) / (1 - r)
Where:
- Sₙ is the sum of the first n terms.
- a is the first term of the sequence.
- r is the common ratio.
- n is the number of terms you want to sum.
Applying the Formula for the First 10 Terms
To find the sum of the first 10 terms (n=10), you need to know the first term (a) and the common ratio (r). Let's walk through the process with an example based on the provided video reference, where although the video does not specify the first term, it implies a first term of 1 and a common ratio (r) of 3.
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Identify the first term (a), common ratio (r), and the number of terms (n):
- Let's assume, a = 1 (as an example).
- The video mentions the r is 3.
- n = 10 (as we are looking for the sum of the first 10 terms)
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Plug the values into the formula:
S₁₀ = 1 * (1 - 3¹⁰) / (1 - 3) -
Calculate 3¹⁰:
- 3¹⁰ = 59049
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Solve the Equation
S₁₀ = (1 - 59049) / (1 - 3)
S₁₀ = (-59048) / (-2)
S₁₀ = 29524
So, the sum of the first 10 terms is 29,524.
Example and Practical Insight
Let's use a different example. Suppose you have a geometric sequence where:
- a = 2
- r = 0.5
- n = 10
Then:
S₁₀ = 2 (1 - 0.5¹⁰) / (1 - 0.5)
S₁₀ = 2 (1 - 0.0009765625) / 0.5
S₁₀ = 2 (0.9990234375) / 0.5
S₁₀ = 2 1.998046875
S₁₀ = 3.99609375
Therefore, the sum of the first 10 terms is 3.99609375
Key Considerations
- Ensure you correctly identify the first term, common ratio, and the desired number of terms.
- If the common ratio is 1, the formula won't work, as you would be dividing by zero. In this case, the sum is simply n * a
- This formula works for both increasing and decreasing geometric sequences.
In summary, to find the sum of the first 10 terms of a geometric sequence, apply the formula, ensuring you know the first term and the common ratio, and make your calculations with care.
