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Geometric Sequences

Discrete Mathematics · Chapter 3: Sequences and SeriesSection 3.4 · lesson 20 of 8710 min · free with a free account

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In this lesson

Multiply by the same amount every step - and watch it grow or decay.

  • Identify a geometric sequence by its common ratio
  • Derive and apply the explicit formula
  • Explain why a recursive definition needs a base case
  • Distinguish exponential growth from decay

Transcript

Everything said in the video, section by section (1,340 words). Read along, or search it.

Sequences and Series

Geometric Sequences

Section three point three built sequences by repeated addition. This section builds them by repeated multiplication. We'll identify a geometric sequence by its common ratio, derive its explicit formula, and see why a recursive definition needs a base case. Then we'll compare growth and decay, in a real engineering trade-off.

The Opening Problem

Exponential Backoff

A mobile app calls a payment API, and the request fails because of a momentary network glitch. Retrying immediately is a bad idea. If the server is briefly overloaded, an instant retry from every failed client only adds to the load. Instead, most production systems use exponential backoff. The wait before each retry grows by a fixed multiple of the previous wait. A typical policy starts at five hundred milliseconds, and doubles after every failed attempt. Five hundred, then one thousand, then two thousand, then four thousand. It's a recommended retry strategy across major cloud software kits. That list of wait times is an instance of a precisely defined object, a geometric sequence.

Definition

The Common Ratio

A sequence is geometric if there's a nonzero constant r, called the common ratio, such that each term is r times the one before it, for every n. Equivalently, r is the ratio of any term to the term immediately before it. And that ratio must be the same constant for every n. That gives a direct test. A sequence is geometric exactly when dividing each term by the one before always gives the same value. It's a genuine if-and-only-if test. A sequence with even one inconsistent ratio isn't geometric, however regular the rest of the pattern looks.

Worked Example 1

Testing the Retry Delays

Let's test the retry delays. Five hundred, one thousand, two thousand, four thousand, eight thousand milliseconds. One thousand over five hundred is two. Two thousand over one thousand is two. Four thousand over two thousand is two. And eight thousand over four thousand is two. Every consecutive ratio equals two. So the sequence is geometric, with first term five hundred and common ratio two.

The Explicit Formula

Unfolding the Definition

Knowing the common ratio lets you generate the sequence one term at a time. But an engineer who needs only the tenth delay shouldn't have to compute the first nine. Unfold the definition. a two is r times a one. a three is r times r a one, which is a one r squared. a four is a one r cubed, and a five is a one r to the fourth. The exponent on r is always one less than the term's index. So the explicit formula is a sub n equals a one times r to the n minus one. It computes any single term directly from its position. A fully general proof, for every n, uses induction, which comes in chapter four.

Worked Example 2

The Tenth Retry Delay

Using a one equal to five hundred and r equal to two, find the tenth delay directly. a ten equals five hundred times two to the ninth. That's five hundred times five hundred twelve, or two hundred fifty-six thousand milliseconds. Two hundred fifty-six seconds. Computing it by repeated multiplication takes n minus one, or nine, multiplications. The work grows in proportion to n. The explicit formula reduces that to a single exponentiation. With an efficient algorithm such as repeated squaring, that takes time that grows only with the logarithm of n. That's a meaningful difference once n is large, and why production code computing one specific term should generally prefer the closed-form expression.

The Recursive Formula

The Rule Alone Isn't Enough

The defining relation can be packaged as a recursive definition. State a one outright, then define each later term as r times the one before. It's worth being precise about what this requires. The rule on its own is a necessary condition. But it isn't sufficient to pin down a unique sequence. With r equal to two, the rule is satisfied by one, two, four, eight. And equally by five hundred, one thousand, two thousand, four thousand. And by every other scaling. Infinitely many sequences satisfy the rule, one for every starting value. The base case supplies the missing information. The rule and the base case together are sufficient to determine the sequence uniquely.

Both Definitions, in Code

Recursive and Explicit Agree

This code implements both definitions for the retry-delay sequence. The recursive version starts with a one, and multiplies by r, once for each later term. The explicit version returns a one times r to the n minus one, in one step. Checked for n from one to ten, the two always agree. Unrolling the recursion by hand for a five gives two times two times two times two times a one. That's sixteen times five hundred, or eight thousand. And the explicit formula gives five hundred times two to the fourth, which is also eight thousand. They must agree, since both describe the same sequence.

Growth Versus Decay

The Ratio Decides

The common ratio decides whether a geometric sequence grows without bound, decays toward zero, or does neither. Assume the first term and the ratio are both positive. If r is greater than one, each ratio exceeds one, so each term is bigger than the last. The sequence strictly increases, and grows without bound. If r is between zero and one, the sequence strictly decreases toward zero. But since r to any finite power is still positive, it never actually reaches zero. Two boundary cases fall outside both. If r equals one, every term equals a one, so the sequence is constant. And if r is negative, the terms alternate in sign. The absolute value of r still decides whether the magnitudes grow or decay, but the sequence isn't monotonic.

Figure 2

A Growing Delay Meets a Shrinking Budget

This figure plots the retry delay against a companion sequence, a shrinking timeout budget. The delay starts at five hundred milliseconds and doubles. The budget starts at eight thousand and halves. The two curves are equal at attempt three, where both reach two thousand milliseconds. After that, the growing delay exceeds the shrinking budget.

Worked Example 4

When Growth Outpaces Decay

Here's why that matters. Suppose the system enforces a shrinking timeout budget, so that repeated retries can't stall the caller forever. Each retry is allotted half the time of the last, starting at eight thousand milliseconds. At attempt three, both sequences equal exactly two thousand. Five hundred times two squared is two thousand, and eight thousand times one half squared is two thousand. Before the third attempt, the budget exceeds the wait the backoff demands. From the third attempt on, the delay exceeds the budget. So a retry would routinely miss its own timeout window. That's precisely why real retry policies cap the maximum delay, or set a floor under the budget, rather than letting either sequence run unchecked. A geometric sequence is a clean mathematical idealization. An engineering policy built on it still needs explicit bounds.

Comparing the Definitions

Equivalent, Not Interchangeable

The two definitions generate identical terms, but they aren't interchangeable in practice. To find one term alone, the recursion must be unrolled through every earlier term. The explicit formula takes one exponentiation. To generate every term in order, both take work in proportion to n, and the recursive form is natural for producing values one at a time. With floating-point numbers, rounding error can compound across the recursion's chained multiplications. In the explicit formula, it's confined to a single call. So a live backoff timer, emitting each value in turn, suits the recursion. A direct question about attempt forty's delay suits the explicit formula.

Takeaways

A Constant Ratio

To summarize. A sequence is geometric exactly when every consecutive ratio equals the same nonzero constant r. The explicit formula computes any term directly from its index. The recursive formula needs both a base case and the rule, since the rule alone fits infinitely many sequences. When r is greater than one, the sequence grows without bound. When r is between zero and one, it decays toward zero, without reaching it. And sums of geometric terms, called geometric series, are a separate question, covered in section three point seven. Here's something to think about. An engineer says it makes no practical difference, which definition a script uses to find the fortieth retry delay. Is that right? Consider the number of operations each approach needs, and what happens when every multiplication carries a small rounding error.

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FAQ

Questions students ask us

About Geometric Sequences

How do you tell if a sequence is geometric?
A sequence is geometric if dividing each term by the one before it always gives the same nonzero constant r, the common ratio. For example, 500, 1000, 2000, 4000, 8000 has ratios of 2 every time. If even one ratio differs, the sequence is not geometric, however regular the rest of the pattern looks.
What is the explicit formula for a geometric sequence and how do you use it?
The explicit formula is a_n = a_1 r^{n-1}, which gives any term directly from its position. It comes from unfolding a_2 = a_1 r, a_3 = a_1 r^2, and so on. For a_1 = 500 and r = 2, the tenth term is 500 \cdot 2^9 = 256000, with no need to compute the first nine terms.
Why does a recursive definition of a geometric sequence need a base case?
The rule a_{n+1} = r a_n alone does not determine one sequence, so a starting value is required. With r = 2, both 1, 2, 4, 8, \dots and 500, 1000, 2000, \dots satisfy the rule. Infinitely many sequences fit, one for each starting value. The base case a_1 together with the rule pins down a unique sequence.
How does the common ratio show whether a geometric sequence grows or decays?
For a positive first term, the size of r decides the behavior. If r > 1, the terms strictly increase without bound (exponential growth). If 0 < r < 1, they strictly decrease toward zero (exponential decay) but never reach it. If r = 1 the sequence is constant, and if r < 0 the signs alternate.

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