Geometric sequence

Definition

A geometric sequence is 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. This property makes geometric sequences distinct from arithmetic sequences, where the difference between consecutive terms is constant. Geometric sequences can model various real-world phenomena, such as population growth, interest calculations, and certain financial models.

5 Must Know Facts For Your Next Test

1. In a geometric sequence, if the first term is 'a' and the common ratio is 'r', the n-th term can be expressed as $$a_n = a imes r^{(n-1)}$$.
2. Geometric sequences can have a common ratio greater than one, which leads to exponential growth, or between zero and one, leading to a decay in values.
3. The sum of the first n terms of a geometric sequence can be calculated using the formula $$S_n = a \frac{1 - r^n}{1 - r}$$ for r not equal to 1.
4. Geometric sequences are widely used in finance for calculating compound interest, where interest is applied to both the principal and accumulated interest over time.
5. An infinite geometric series converges to a finite value if the absolute value of the common ratio is less than one, and can be calculated using $$S = \frac{a}{1 - r}$$.

Review Questions

• How do you determine whether a given sequence is geometric?
  • To determine if a sequence is geometric, check if there is a constant ratio between consecutive terms. Calculate the ratio by dividing each term by its previous term. If this ratio remains constant throughout the sequence, it confirms that it is indeed geometric. If at any point the ratio changes, then it is not a geometric sequence.
• Describe how geometric sequences can be applied in real-world scenarios, particularly in finance.
  • Geometric sequences are applicable in finance mainly for calculating compound interest. When interest is compounded, each period's interest is calculated based on the total amount including previously earned interest. This forms a geometric sequence where the principal amount grows exponentially over time based on a fixed interest rate, allowing investors to predict future earnings effectively.
• Evaluate how understanding geometric sequences enhances your ability to analyze patterns in natural phenomena such as population growth.
  • Understanding geometric sequences allows you to analyze and model situations where growth occurs at an increasing rate, such as population growth under ideal conditions. By recognizing that population increases can often be modeled with geometric sequences due to factors like birth rates and resource availability, you can predict future populations using mathematical formulas derived from these sequences. This analytical skill enables better planning and resource management in fields like ecology and urban development.

Related terms

common ratio: The fixed number that each term in a geometric sequence is multiplied by to obtain the next term.

exponential growth: A rapid increase in a quantity where the rate of growth is proportional to the current amount, often represented by a geometric sequence.

infinite geometric series: A sum of an infinite number of terms of a geometric sequence that converges to a finite value if the absolute value of the common ratio is less than one.