Arithmetic & Geometric Series
数列とは規則に従って並んだ数の列で、級数はその和です。この単元では二つの基本数列——一定の差で増える等差数列と、一定の比で増える等比数列——の一般項と和の公式を扱います。さらに等比数列では、公比の絶対値が より小さいとき項を無限に足しても有限の値に収束する「無限和」も学びます。公式の暗記だけでなく、初項 ・公差 ・公比 を問題文から正しく読み取る力が AS Pure 1 で問われます。
A sequence is a list of numbers following a rule, and a series is the sum of such a list. This topic covers two fundamental sequences — the arithmetic progression, which grows by a constant difference, and the geometric progression, which grows by a constant ratio — together with formulas for their th terms and sums. For geometric progressions you will also meet the sum to infinity, where adding infinitely many terms still gives a finite value provided the common ratio has absolute value less than . AS Pure 1 tests not just recall of the formulas but the skill of reading the first term , common difference and common ratio correctly from the question.
The nth term of an arithmetic progression
等差数列（AP）は、隣り合う項の差が一定の数列です。その一定の差を公差 、最初の項を初項 とよびます。第 項は初項に公差を 回足したものなので、次の式で表されます。 なら増加、 なら減少する数列です。
An arithmetic progression (AP) is a sequence in which the difference between consecutive terms is constant. That constant is the common difference , and the starting value is the first term . The th term is the first term with the common difference added times, giving the formula below. If the sequence increases; if it decreases.
The sum of an arithmetic progression
初項から第 項までの和 は、項を前後で対にして足すという有名な発想から導けます。次の二つは同じ式で、最終項 が分かっているときは右の形が便利です。「初項と末項の平均に項数を掛ける」と覚えると速いです。
The sum of the first terms can be derived from the famous idea of pairing terms from each end. The two forms below are equivalent; the right-hand version is handy when the last term is known. A quick way to remember it: the average of the first and last term, multiplied by the number of terms.
The nth term of a geometric progression
等比数列（GP）は、隣り合う項の比が一定の数列です。その一定の比を公比 とよびます。第 項は初項に公比を 回掛けたものなので、次の式になります。 が負の場合は項の符号が交互に変わる「交代数列」になる点に注意してください。
A geometric progression (GP) is a sequence in which the ratio between consecutive terms is constant. That constant is the common ratio . The th term is the first term multiplied by the common ratio times, giving the formula below. Note that if is negative the terms alternate in sign, producing an alternating sequence.
The sum of a geometric progression
初項から第 項までの和は次の公式で求めます。 のときは左、 のときは符号を整理した右の形が計算しやすいです（ が前提）。導出は を作ると途中項がすべて消えることから分かります。
The sum of the first terms is given by the formula below. When the left form is convenient, and when the right form (with signs rearranged) is tidier; both require . The derivation comes from forming , after which all the intermediate terms cancel.
The sum to infinity of a geometric series
公比の絶対値が より小さい（）とき、 を大きくすると は に近づきます。すると は有限の値に収束し、その極限を無限和 とよびます。 のときは発散して値をもたない点が試験では頻出の注意事項です。
When the common ratio satisfies , the power approaches as grows large. The sum then converges to a finite value, and this limit is called the sum to infinity . A point examiners love to test: if the series diverges and has no sum to infinity.
Worked examples
Practice
南数塾は、Cambridge Primary / IGCSE / A-Level の各単元を、日本式の分かりやすさと英語の数学用語の両面から指導します。無料体験からどうぞ。
