Geometric Sequence

The Core Concept

A geometric sequence is a list of numbers where each term is found by multiplying the previous term by a constant called the common ratio ($r$).

$$a_n = a_{n-1} \cdot r$$

The recursive formula defines a term based on its immediate predecessor.

$a_n$=Current term(unitless)

$a_{n-1}$=Previous term(unitless)

$r$=Common ratio(multiplier)

$r > 1$

→

The sequence grows indefinitely (Growth)

$0 < r < 1$

→

The sequence shrinks toward zero (Decay)

$r < 0$

→

The sequence alternates in sign

Common Ratio: Found by dividing any term by its predecessor ($r = a_n / a_{n-1}$).

Recursive Formula: Defines the relationship between adjacent terms.

Initial Term: The starting value ($a_1$) which determines the scale of the entire sequence.

Zero Terms Forbidden: No term of a G.P. can be zero, since $r = a_n / a_{n-1}$ is undefined when $a_{n-1} = 0$.

Sign Pattern: When $r < 0$, terms alternate sign — even-indexed powers of $r$ are positive, odd-indexed powers are negative.

Finding Any Term

The explicit formula allows you to calculate the $n^{th}$ term directly without knowing all previous values by applying the common ratio repeatedly.

$$a_n = a_1 r^{n-1}$$

Directly links term position to its value using powers of the ratio.

$a_n$=The nth term value(unitless)

$a_1$=First term(unitless)

$r$=Common ratio(unitless)

$n$=Position index(integer)

$n = 1$

→

$a_1 r^0 = a_1$ (Consistent with initial term)

Exponent: The power is $n-1$ because the first term involves zero multiplications of $r$.

Position Index: Always a positive integer representing the term's place in line.

Finding $a_1$ and $r$ from Two Terms: Divide the later term by the earlier term to eliminate $a_1$: $\frac{a_m}{a_k} = r^{m-k}$, then solve for $r$.

Proportionality: $a_n \propto r^n$ — doubling $r$ raises every term exponentially, while doubling $a_1$ simply scales the entire sequence.

Summation of Terms

A finite geometric series is the sum of a specific number of terms in a sequence, calculated using a formula that accounts for the cumulative multiplicative effect.

$$S_n = \frac{a_1(1 - r^n)}{1 - r}$$

Calculates the total of the first $n$ terms efficiently.

$S_n$=Partial sum of n terms(unitless)

$a_1$=First term(unitless)

$r$=Common ratio(unitless)

$r = 1$

→

$S_n = n \cdot a_1$ (Arithmetic-like sum)

Partial Sum: The total value up to a certain point $n$.

Alternative Form: When $|r| > 1$, use $S_n = \frac{a_1(r^n - 1)}{r - 1}$ to avoid negative numerator and denominator.

Derivation Trick: Multiply $S_n$ by $r$, then subtract: $S_n - rS_n = a_1 - a_1 r^n$. This telescoping cancellation is why the formula works.

Sigma Notation: Compact way to write sums using $\sum_{k=1}^{n} a_1 r^{k-1}$.

Infinite Sums & Convergence

If $|r| < 1$, an infinite geometric series will converge to a specific limit, meaning the sum of infinitely many terms is a finite number.

$$S_\infty = \frac{a_1}{1 - r}, \text{ for } |r| < 1$$

Determines the value an infinite series approaches as $n \to \infty$.

$S_\infty$=Sum to infinity(unitless)

$a_1$=Initial term(unitless)

$r$=Common ratio(unitless)

$|r| \ge 1$

→

The series diverges (Sum approaches $\pm \infty$ or oscillates)

Convergence: The behavior of a series that approaches a finite limit.

Divergence: Occurs when terms do not shrink fast enough ($|r| \ge 1$) to have a finite sum.

Oscillatory Case: When $r = -1$, the partial sums alternate between $a_1$ and $0$ — the series neither converges nor diverges to infinity.

Recurring Decimals: Every repeating decimal is an infinite G.P. For example, $0.\overline{23} = \frac{23}{100} + \frac{23}{10000} + \cdots$ with $r = \frac{1}{100}$, giving $\frac{23}{99}$.

Derivation from Finite: $S_\infty$ is obtained from $S_n$ by noting $r^n \to 0$ as $n \to \infty$ when $|r| < 1$, so $1 - r^n \to 1$.

Geometric Mean

A number $G$ is the geometric mean between two numbers $a$ and $b$ if $a, G, b$ form a G.P. This gives $G^2 = ab$, so $G = \pm\sqrt{ab}$.

$$G = \pm\sqrt{ab}$$

The geometric mean is the square root of the product of two numbers, preserving the multiplicative relationship.

$G$=Geometric mean between $a$ and $b$(unitless)

$a, b$=The two given numbers(unitless)

$a = b$

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$G = \pm a$ — the geometric mean of equal numbers is the number itself

$a$ and $b$ have opposite signs

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$ab < 0$, so $\sqrt{ab}$ is imaginary — real G.M. does not exist

Derivation: If $a, G, b$ are in G.P., then $\frac{G}{a} = \frac{b}{G}$, which gives $G^2 = ab$.

Two G.P.s Result: $G = +\sqrt{ab}$ gives ratio $r > 0$; $G = -\sqrt{ab}$ gives ratio $r < 0$. Both are valid geometric sequences.

AM ≥ GM Inequality: For positive $a, b$: $\frac{a+b}{2} \ge \sqrt{ab}$, with equality only when $a = b$.

To insert $n$ geometric means between $a$ and $b$, we form a G.P. of $n+2$ terms: $a, G_1, G_2, \ldots, G_n, b$. The common ratio is found from $b = ar^{n+1}$.

$$r = \left(\frac{b}{a}\right)^{\frac{1}{n+1}}$$

The common ratio for inserting $n$ geometric means creates equally-spaced multiplicative steps between $a$ and $b$.

$r$=Common ratio of the resulting G.P.(unitless)

$n$=Number of geometric means to insert(integer)

$a, b$=The boundary values(unitless)

$n = 1$

→

$r = \sqrt{b/a}$, recovering the single geometric mean formula

Total Terms: The sequence has $n + 2$ terms (including $a$ and $b$ at the ends).

$k$-th Mean: $G_k = a \cdot r^k = a\left(\frac{b}{a}\right)^{k/(n+1)}$ for $k = 1, 2, \ldots, n$.

Product Property: The product of all $n$ geometric means equals $(\sqrt{ab})^n$, and their geometric mean equals $\sqrt{ab}$.