General Term Formula of Geometric Sequences

The general term formula is the core tool of geometric sequences. It allows us to directly find any term of the sequence, reflecting the mathematical essence of exponential growth.

Derivation of the Formula

For a geometric sequence $\{a_n\}$, let the first term be $a_1$ and the common ratio be $q$. Let us derive the expression for the $n$th term $a_n$.

According to the definition of a geometric sequence:

$$\begin{aligned} a_2 &= a_1 \cdot q \\ a_3 &= a_2 \cdot q = a_1 \cdot q^2 \\ a_4 &= a_3 \cdot q = a_1 \cdot q^3 \\ a_5 &= a_4 \cdot q = a_1 \cdot q^4 \\ &\vdots \end{aligned}$$

Observe the pattern: from $a_1$ to $a_n$, we need to multiply by the common ratio $q$ $(n-1)$ times.

Variations of the General Term Formula

The general term formula can also be written in other forms:

Form 1: Based on an Arbitrary Term

If the $m$th term $a_m$ is known, then:

$a_n = a_m \cdot q^{n-m}$

This formula is especially useful when a certain term (not necessarily the first) is known.

Form 2: The Exponential Function Form

Rewriting the general term formula:

$a_n = a_1 \cdot q^{n-1} = \frac{a_1}{q} \cdot q^n$

Let $c = \frac{a_1}{q}$. Then:

$a_n = c \cdot q^n$

This shows that the general term formula of a geometric sequence is essentially an exponential function (when $q > 0$ and $q \neq 1$).

Worked Examples

Example 1: Given the first term and common ratio, find a certain term

In the geometric sequence $\{a_n\}$, the first term is $a_1 = 3$ and the common ratio is $q = 2$. Find the 8th term $a_8$.

Solution:

$a_8 = a_1 \cdot q^{8-1} = 3 \cdot 2^7 = 3 \cdot 128 = 384$

Example 2: Given two terms, find the general term formula

In the geometric sequence $\{a_n\}$, $a_3 = 12$ and $a_6 = 96$. Find the general term formula.

Solution:

Using the general term formula:

$$\begin{cases} a_3 = a_1 \cdot q^2 = 12 \\ a_6 = a_1 \cdot q^5 = 96 \end{cases}$$

Dividing the two equations: $\frac{a_6}{a_3} = \frac{a_1 \cdot q^5}{a_1 \cdot q^2} = q^3 = \frac{96}{12} = 8$

So $q^3 = 8$, giving $q = 2$.

Substituting into the first equation: $a_1 \cdot 2^2 = 12 \quad \Rightarrow \quad a_1 = 3$

Therefore, the general term formula is: $a_n = 3 \cdot 2^{n-1}$

Example 3: Compound Interest Calculation

A person deposits 10000 yuan in a bank at an annual interest rate of 5% with compound interest. What is the amount after $n$ years?

Solution:

This is a geometric sequence problem:

• First term (principal): $a_1 = 10000$ yuan
• Common ratio: $q = 1 + 5\% = 1.05$
• The $(n+1)$th term is the amount after $n$ years

$a_{n+1} = 10000 \cdot 1.05^n$

For example, the amount after 10 years: $a_{11} = 10000 \cdot 1.05^{10} \approx 16289 \text{ yuan}$

Practice Problems

Exercise 1

In the geometric sequence $\{a_n\}$, $a_1 = 2$ and $q = 3$. Find $a_6$.

Exercise 2

In the geometric sequence $\{a_n\}$, $a_2 = 4$ and $a_5 = 32$. Find the first term $a_1$ and the common ratio $q$.

Exercise 3

In the geometric sequence $\{a_n\}$, $a_1 = 3$ and $q = \frac{1}{2}$. From which term does the sequence become less than $\frac{1}{100}$?

Exercise 4

Adapted from a postgraduate entrance examination problem

The geometric sequence $\{a_n\}$ satisfies $a_1 + a_2 = 3$ and $a_2 + a_3 = 6$. Find $a_1$ and $q$.

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