Arithmetic and Geometric Sequences

date2026-07-14document_iddoc_2e609f0db64f8a2e725977a907f28c82description等差数列と等比数列を漸化式で定義し、一般項と有限和を端点・例外を含めて証明する二言語講義。prerequisites数列の基本type講義content_typelecturestatusactive

mathhighschoolsequencelecture

1Introduction

The central task of this lecture is to distinguish additive change by a constant amount from multiplicative change by a constant factor. We define both classes by recurrences between adjacent terms and prove their general-term and finite-sum formulas.

2Definitions

Consider a real sequence $(a_{n})_{n \geq 1}$ . It is an {arithmetic sequence} if there is a constant $d \in R$ such that

a_{n + 1} = a_{n} + d

for every $n \geq 1$ . The number $d$ is its {common difference}. This definition includes constant sequences, for which $d = 0$ .

It is a {geometric sequence} if there is a constant $r \in R$ such that

a_{n + 1} = r a_{n}

for every $n \geq 1$ . The number $r$ is a {common ratio}. This definition includes the zero sequence arising from $a_{1} = 0$ . For the zero sequence, every $r$ satisfies the recurrence, so the common ratio need not be unique.

3General Terms

3.1Arithmetic Sequences

For every $n \geq 1$ ,

[PARSE ERROR: Undefined("Command(\"boxed\")")] a_{n} = a_{1} + (n - 1) d .

At $n = 1$ , the right-hand side is $a_{1}$ . For $n \geq 2$ , sum $a_{k + 1} - a_{k} = d$ over $k = 1, \dots, n - 1$ :

a_{n} - a_{1} = \sum_{k = 1}^{n - 1} (a_{k + 1} - a_{k}) = (n - 1) d .

The sum in the middle telescopes because its intermediate terms cancel, proving the formula.

3.2Geometric Sequences

For every $n \geq 1$ ,

[PARSE ERROR: Undefined("Command(\"boxed\")")] a_{n} = a_{1} r^{n - 1} .

At $n = 1$ , this follows from $r^{0} = 1$ . If $a_{n} = a_{1} r^{n - 1}$ , then $a_{n + 1} = r a_{n} = a_{1} r^{n}$ ; mathematical induction proves the formula. This proof uses no division and remains valid when $a_{1} = 0$ or $r = 0$ .

4Finite Sums

For $1 \leq m \leq N$ , define

S_{m, N} := \sum_{n = m}^{N} a_{n} .

There are $N - m + 1$ terms.

4.1Finite Arithmetic Sums

[PARSE ERROR: Undefined("Command(\"boxed\")")] S_{m, N} = \frac{N - m + 1}{2} (a_{m} + a_{N}) .

Add the same sum written in reverse order:

2 S_{m, N} = \sum_{j = 0}^{N - m} (a_{m + j} + a_{N - j}) .

The arithmetic property makes every parenthesis equal to $a_{m} + a_{N}$ , and there are $N - m + 1$ parentheses. This proves the formula, including the endpoint case $m = N$ , where it gives $a_{m}$ .

In particular, for $m = 1$ ,

[PARSE ERROR: Undefined("Command(\"boxed\")")] \sum_{n = 1}^{N} a_{n} = \frac{N}{2} {2 a_{1} + (N - 1) d} = \frac{N}{2} (a_{1} + a_{N}) .

4.2Finite Geometric Sums

If $r \neq 1$ , then

[PARSE ERROR: Undefined("Command(\"boxed\")")] S_{m, N} = a_{m} \frac{1 - r^{N - m + 1}}{1 - r} .

Write $S_{m, N} = a_{m} + a_{m} r + \dots + a_{m} r^{N - m}$ and subtract $r S_{m, N}$ to obtain

(1 - r) S_{m, N} = a_{m} - a_{m} r^{N - m + 1} .

Division by $1 - r \neq 0$ proves the formula, including $m = N$ . If $r = 1$ , that division is invalid and every term equals $a_{m}$ , so instead

[PARSE ERROR: Undefined("Command(\"boxed\")")] S_{m, N} = (N - m + 1) a_{m} .

Both formulas include $a_{m} = 0$ .

5Tests by Differences and Ratios

If $a_{n + 1} - a_{n} = d$ is the same constant for every $n \geq 1$ , the sequence is arithmetic by definition.
If $a_{n} \neq 0$ for every $n \geq 1$ and $a_{n + 1} / a_{n} = r$ is the same constant, then $a_{n + 1} = r a_{n}$ , so the sequence is geometric.
A ratio test cannot be applied where its denominator $a_{n}$ is zero. When zero terms may occur, test $a_{n + 1} = r a_{n}$ directly for a single constant $r$ , without division.
Agreement among finitely many displayed differences or ratios does not classify an infinite sequence unless the same rule is guaranteed for every unshown term.

For example, $2, 5, 8, 11, \dots$ has common difference $3$ and $a_{n} = 3 n - 1$ . The nonzero sequence $5, 10, 20, 40, \dots$ has ratio $2$ and $a_{n} = 5 \cdot 2^{n - 1}$ . By contrast, $0, 0, 0, \dots$ is geometric, but the ratio test is unavailable because $0 / 0$ is undefined.

6Structural Interpretation

The arithmetic general term $a_{1} + (n - 1) d$ is affine in the index and is a discrete counterpart of an affine function. When $r > 0$ , a geometric sequence consists of the values of the exponential-form function $x \mapsto a_{1} r^{x - 1}$ at integer points. When $r < 0$ and $a_{1} \neq 0$ , its signs alternate, while $r = 0$ makes every term after the first zero, so these cases must be distinguished from that correspondence. Moreover, a first-order recurrence $a_{n + 1} = p a_{n} + q$ is arithmetic when $p = 1$ ; when $p \neq 1$ , its deviation from an equilibrium point is geometric.

7Scope

The finite-sum identities in this lecture concern finitely many terms and require no convergence assumption. Summing an infinite geometric series requires additional conditions such as $| r | < 1$ . The formulas here do not apply directly when the difference or multiplier depends on the index.

8Basic Formulas

[PARSE ERROR: Undefined("Command(\"boxed\")")] a_{n + 1} = a_{n} + d ⟹ a_{n} = a_{1} + (n - 1) d

[PARSE ERROR: Undefined("Command(\"boxed\")")] a_{n + 1} = r a_{n} ⟹ a_{n} = a_{1} r^{n - 1}

[PARSE ERROR: Undefined("Command(\"boxed\")")] \sum_{n = m}^{N} a_{n} = \frac{N - m + 1}{2} (a_{m} + a_{N}) a r i t h m e t i c) [PARSE ERROR: Undefined("RBrace")]

\boxed{\sum_{n=m}^{N}a_n=\begin{cases}a_m\dfrac{1-r^{N-m+1}}{1-r},&r\ne1,\\(N-m+1)a_m,&r=1\end{cases}\quad\text{(geometric)}}\boxed{\sum_{n=m}^{N}a_n=\begin{cases}a_m\dfrac{1-r^{N-m+1}}{1-r},&r\ne1,\\(N-m+1)a_m,&r=1\end{cases}\quad\text{(geometric)}}

9Summary

Arithmetic sequences are defined by adding a constant amount; geometric sequences are defined by multiplying by a constant factor.
General-term and finite-sum formulas must track the first term, last term, and number of terms; a geometric sum must treat $r = 1$ separately.
A ratio test requires nonzero denominators. When zero terms occur, test the recurrence directly.

10Related Lectures

data/lecture/math/sequence/sequences-portal.lecture.n.md data/lecture/math/sequence/sequences-and-recurrences.lecture.n.md data/lecture/math/sequence/first-order-recurrences.lecture.n.md data/lecture/math/sequence/limits-of-sequences.lecture.n.md data/lecture/math/sequence/series-basics.lecture.n.md