What is a Geometric Series?
A geometric series is the sum of an infinite or finite geometric sequence of numbers. A geometric sequence is one in which the ratio of consecutive terms is constant, known as the common ratio (r).
Geometric Series Sum Formula
To find the sum of a finite geometric series with n terms, we use the following formula:
- Sₙ is the sum of the first n terms
- a₁ is the first term
- r is the common ratio
- n is the number of terms
If r = 1, the sum is simply a₁ × n since all terms are equal to a₁.
Applications of Geometric Series
Geometric series calculations are not just for the classroom, but are widely used in real life. For example:
- การเงินและการลงทุน: Calculating future value of compound interest savings, mortgage payments, and present value of cash flows.
- วิทยาศาสตร์: Describing radioactive decay where the amount halves over fixed time intervals.
- การแพร่กระจายของข้อมูลและไวรัส: Network expansion or viral spread where one entity infects a fixed number of others.
Finite vs Infinite Geometric Series
A finite geometric series always has an end. For an infinite geometric series, the sum converges to a finite value only if the absolute value of r is less than 1 (|r| < 1). The formula for an infinite geometric series is S = a₁ / (1 - r).
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