Sum formula for geometric series

Definition

The sum formula for a geometric series is a mathematical expression used to calculate the total sum of the terms in a geometric sequence. This formula is significant because it helps in determining the sum when each term is produced by multiplying the previous term by a constant factor, known as the common ratio. Understanding this formula is essential when working with series and summation notation as it allows for efficient calculations of infinite or finite sums.

5 Must Know Facts For Your Next Test

1. The sum formula for a finite geometric series is given by $$S_n = a \frac{1 - r^n}{1 - r}$$, where 'a' is the first term, 'r' is the common ratio, and 'n' is the number of terms.
2. For an infinite geometric series with a common ratio \( |r| < 1 \), the sum can be calculated using $$S = \frac{a}{1 - r}$$.
3. If the common ratio 'r' equals 1 in a geometric series, the formula simplifies to just 'na', which means you are simply adding 'a' repeatedly 'n' times.
4. Understanding how to manipulate the sum formula for geometric series can help solve problems involving finance, physics, and computer science where growth patterns are exponential.
5. The convergence of an infinite geometric series is dependent on the absolute value of the common ratio being less than one, otherwise, the series diverges.

Review Questions

• How would you derive the sum formula for a finite geometric series from its definition?
  • To derive the sum formula for a finite geometric series, start with the definition of the sum: $$S_n = a + ar + ar^2 + ... + ar^{n-1}$$. If you multiply both sides by the common ratio 'r', you get $$rS_n = ar + ar^2 + ... + ar^n$$. Subtracting these two equations eliminates all but the first and last terms, leading to $$S_n(1 - r) = a(1 - r^n)$$. Rearranging gives you $$S_n = a \frac{1 - r^n}{1 - r}$$.
• Explain how to apply the sum formula for an infinite geometric series and under what conditions it can be used.
  • To apply the sum formula for an infinite geometric series, you use $$S = \frac{a}{1 - r}$$ when the absolute value of the common ratio 'r' is less than 1. This condition ensures that as you continue adding terms, they become smaller and smaller, allowing the series to converge to a finite value. If 'r' is equal to or greater than 1, then the series does not converge and instead diverges towards infinity.
• Critically analyze how understanding the sum formula for geometric series impacts problem-solving in real-world applications.
  • Understanding the sum formula for geometric series greatly enhances problem-solving skills in various real-world contexts such as finance and population studies. For instance, in finance, calculating compound interest involves recognizing that investments grow in a geometric pattern over time. By applying this formula effectively, one can determine future values based on initial investments and growth rates. Furthermore, recognizing when to use this formula allows for quick calculations in scenarios involving exponential growth or decay, leading to more informed decision-making.

Related terms

Geometric Sequence: A sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

Common Ratio: The constant factor between consecutive terms in a geometric sequence, determined by dividing any term by the previous term.

Finite Series: A sum of a limited number of terms from a sequence, which can be calculated using specific formulas depending on the nature of the series.