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.
Understanding Math: Geometric Sequences
A useful way to recognize this pattern is to compare neighboring terms by division. Subtraction can be misleading because the gaps usually change. For example, in the list five, fifteen, forty five, one hundred thirty five, the differences are ten, thirty, and ninety.
They are not constant. Each new term is three times the term before it, so the multiplier remains constant. This is the key distinction between geometric sequences and arithmetic sequences, where a fixed amount is added each time.
Always check several pairs of terms. One matching pair alone does not prove a whole list follows one rule.
The value of the multiplier tells a detailed story about the sequence. A positive multiplier greater than one produces increasing positive terms. A positive multiplier between zero and one produces values that get closer to zero without reaching it.
A negative multiplier makes signs switch from positive to negative on every step. Its size still controls growth or shrinkage. A multiplier of negative two makes magnitudes double, while a multiplier of negative one half makes magnitudes halve.
A multiplier of one leaves every term unchanged. A multiplier of zero creates a special case. After the first multiplication, every later term is zero, so division cannot be used to find a ratio from those zero terms.
Powers explain why these sequences change so quickly. Repeated multiplication by the same number builds a power. If a quantity doubles every year, it is not simply two times larger after many years.
After ten yearly doublings, it is the starting amount multiplied by two ten times. This is why compound interest can grow surprisingly fast when interest stays invested. The same reasoning appears in biology when a simplified population model has a fixed growth factor per generation.
Real populations rarely keep one exact factor forever because food, space, disease, and weather impose limits. The sequence is still a useful first model over a limited time.
Geometric sums track the total accumulated amount rather than one term. Suppose a ball travels half as far after each bounce. The distances form a shrinking pattern.
Adding every bounce creates a total that approaches a fixed limit, even though there could be endlessly many smaller bounces in the mathematical model. This only works when the size of the multiplier is less than one. When its size is one or greater, later terms do not become small enough for an infinite total to settle.
When solving sum problems, identify whether the question asks for a particular term or for the total of many terms. Also count terms carefully.
The first term is often numbered one, so the number of multiplications is one less than the term number. That small indexing detail causes many otherwise correct answers to be off by one.
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.