Construct and interpret geometric sequences with explicit and recursive rules, and connect them to exponential functions whose domain is the integers (A.FGR, Functional and Graphical Reasoning).

What this topic is asking

This Functional and Graphical Reasoning (A.FGR) standard covers geometric sequences and frames them as a special kind of exponential function. A geometric sequence multiplies by a fixed amount, the common ratio, to get from one term to the next, in contrast to the arithmetic sequence's constant addition. The Georgia Milestones EOC asks for the explicit rule for the $n$th term, the recursive rule, and a specified term. As with arithmetic sequences, the formulas are usually not on the reference sheet, so memorize them, and the connection to exponential functions is the conceptual reward.

The common ratio

In a geometric sequence, consecutive terms have a constant ratio $r$. Find it by dividing any term by the previous one: for $3, 6, 12, 24, \dots$, $r = \frac{6}{3} = 2$. A ratio between 0 and 1 gives a decreasing sequence, like $64, 32, 16, \dots$ with $r = \frac{1}{2}$.

The explicit rule

The explicit rule gives the $n$th term directly.

The exponent is $(n - 1)$ because the first term has been multiplied by $r$ zero times. For $3, 6, 12, \dots$: $a_n = 3(2)^{n-1}$, so $a_6 = 3(2)^5 = 96$.

The recursive rule

The recursive rule defines each term from the previous one and includes the starting value.

For the same sequence: $a_n = 2a_{n-1}$ with $a_1 = 3$.

A geometric sequence is an exponential function

The explicit rule $a_n = a_1 \cdot r^{\,n-1}$ is exponential in $n$: it is a constant times a base raised to a power. So a geometric sequence is an exponential function whose domain is the positive integers. The common ratio is the base, and a ratio greater than 1 grows while a ratio between 0 and 1 decays, exactly like an exponential model.

How the Milestones examines this topic

• Multiple choice. Choose the explicit rule, with the $(n - 1)$-versus-$n$ trap and an arithmetic-rule distractor.
• Numeric entry. Find a specified term or the common ratio.
• Constructed response. Write both rules and find a term, stating the starting value.

Why geometric sequences explode or vanish

The behavior of a geometric sequence is far more dramatic than an arithmetic one, and seeing why helps you predict and check answers. Because each term multiplies by $r$, the effect compounds: with $r = 2$ the terms double every step ($3, 6, 12, 24, 48, \dots$), so they grow without bound very quickly, much faster than any arithmetic sequence eventually. With $0 < r < 1$, each term is a fraction of the last ($64, 32, 16, 8, \dots$), so the terms shrink toward zero but never reach it. This is the same runaway-growth-or-decay behavior as an exponential function, which is exactly the point of pairing them. On the EOC, if a sequence's terms are changing by a constant factor rather than a constant amount, expect geometric and reach for $a_1 r^{\,n-1}$, and expect the values to grow or shrink fast.

Modeling with geometric sequences

Geometric sequences model repeated proportional change: a bacteria count doubling each hour, a bouncing ball reaching a fixed fraction of its previous height, a savings balance multiplied by a growth factor each period. Writing the model means finding the first term and the common ratio (the multiplier per step), then using the explicit rule for "how much after $n$ steps." This is the discrete cousin of the exponential growth and decay models, so the same "what is the multiplier per period" reasoning applies, and recognizing the shared structure lets you carry exponential intuition into sequence problems.

Try this

Q1. Write the explicit rule for $5, 15, 45, \dots$ and find $a_4$. [2 points]

• Cue. $r = 3$, $a_n = 5(3)^{n-1}$; $a_4 = 5(3)^3 = 135$.

Q2. A sequence has $a_1 = 1000$ and $r = \frac{1}{10}$. Write the recursive rule. [1 point]

• Cue. $a_n = \frac{1}{10}a_{n-1}$ with $a_1 = 1000$.