Sum Formula

Definition

The sum formula is a mathematical expression used to calculate the total of the terms in a sequence, particularly in the context of geometric sequences. It provides a way to determine the sum of a series of numbers where each term is derived by multiplying the previous term by a constant ratio. This formula becomes essential when dealing with finite or infinite series, allowing one to find the total without needing to add each term individually.

5 Must Know Facts For Your Next Test

1. The formula for the sum of the first 'n' terms of a geometric sequence can be expressed as $$S_n = a \frac{1 - r^n}{1 - r}$$, where 'a' is the first term and 'r' is the common ratio.
2. For an infinite geometric series where the absolute value of the common ratio is less than 1, the sum converges and is calculated using $$S = \frac{a}{1 - r}$$.
3. The sum formula simplifies calculations significantly by avoiding the need to individually add up each term in a geometric sequence.
4. If the common ratio 'r' is greater than or equal to 1, the series does not converge, meaning it grows indefinitely.
5. Understanding the sum formula helps in real-world applications like calculating interest in finance and understanding exponential growth patterns.

Review Questions

• How does the sum formula for a geometric series differ when considering finite versus infinite series?
  • The sum formula for a finite geometric series allows you to calculate the total of a specific number of terms using $$S_n = a \frac{1 - r^n}{1 - r}$$. In contrast, for an infinite geometric series with a common ratio whose absolute value is less than 1, the total converges to a finite value calculated by $$S = \frac{a}{1 - r}$$. This difference highlights how series behave differently based on their properties and helps to inform their applications.
• Given a geometric sequence with a first term of 3 and a common ratio of 2, calculate the sum of the first 5 terms.
  • To find the sum of the first 5 terms, we can use the finite sum formula: $$S_n = 3 \frac{1 - 2^5}{1 - 2}$$. Calculating this gives us: $$S_5 = 3 \frac{1 - 32}{-1} = 3 \times 31 = 93$$. Thus, the sum of the first 5 terms in this sequence is 93.
• Evaluate how understanding the sum formula impacts financial decisions related to investments involving compound interest.
  • Understanding the sum formula allows individuals to calculate future values of investments with compound interest more accurately. For example, when using an infinite geometric series model to represent continuous compounding, investors can determine how much their investment will grow over time with varying interest rates. This knowledge aids in making informed decisions about saving and investing money, as it translates complex calculations into manageable formulas that predict growth and returns effectively.

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 its preceding term.

Finite Series: The sum of the terms of a finite sequence, which has a specific number of terms that can be counted.