A geometric sequence is a list of numbers in which each term is found by multiplying the previous term by the same constant. That constant is called the common ratio, and it controls whether the sequence grows, shrinks, alternates signs, or stays the same. Geometric sequences matter because they model repeated percent change, compound interest, population growth, radioactive decay, and many scaling patterns in science and math.
Key Facts
- Common ratio: r = a_n / a_(n-1), as long as a_(n-1) is not 0.
- Explicit formula: a_n = a_1 r^(n - 1).
- Recursive formula: a_n = r a_(n-1), with starting value a_1.
- Finite geometric sum: S_n = a_1(1 - r^n) / (1 - r), for r not equal to 1.
- Infinite geometric sum: S_infinity = a_1 / (1 - r), only when |r| < 1.
- For the sequence 3, 6, 12, 24, 48, ..., the common ratio is r = 2 and a_n = 3(2)^(n - 1).
Vocabulary
- Geometric sequence
- A sequence in which each term is produced by multiplying the previous term by a constant common ratio.
- Common ratio
- The constant multiplier r found by dividing any term by the previous nonzero term.
- Explicit formula
- A formula that gives a_n directly from the term number n without listing earlier terms.
- Finite geometric series
- The sum of a fixed number of terms from a geometric sequence.
- Convergent infinite series
- An infinite sum that approaches a finite value, which happens for a geometric series when the absolute value of the common ratio is less than 1.
Common Mistakes to Avoid
- Adding a constant difference instead of multiplying by a common ratio. This turns the pattern into an arithmetic sequence, not a geometric sequence.
- Using a_n = a_1 r^n instead of a_n = a_1 r^(n - 1). The exponent is n - 1 because the first term has not been multiplied by r yet.
- Applying the infinite sum formula when |r| is greater than or equal to 1. The infinite geometric sum only has a finite value when the terms shrink toward 0.
- Forgetting that a negative common ratio makes signs alternate. The magnitude may grow or shrink, but the sign pattern is also part of the sequence.
Practice Questions
- 1 For the geometric sequence 3, 6, 12, 24, 48, ..., find the common ratio, write the explicit formula, and calculate a_8.
- 2 A geometric sequence has a_1 = 80 and r = 1/2. Find the first 6 terms and the sum S_6.
- 3 A ball rebounds to 60% of its previous height after each bounce. If it is dropped from 10 meters, explain why the rebound heights form a geometric sequence and whether the total rebound distance has a finite limit.