--- title: Common Difference/Ratio & General Term of Sequences description: Learn to find the common difference (arithmetic) or ratio (geometric) and calculate the general (n-th) term of a sequence. Includes examples. url: https://astra-ai.co/videos/sequences/arithmetic-sequence/difference-and-calculation-of-general-term/ locale: en type: video-lesson published: 2026-10-01T09:54:24.081Z updated: 2026-10-01T09:54:24.081Z alternates: it: https://astra-ai.co/it/videos/successioni/successione-aritmetica/differenza-calcolo-termine-generale/ lt: https://astra-ai.co/lt/videos/sekos/aritmetine-seka/pamoka-2c1c6955/ th: https://astra-ai.co/th/videos/lamdap/bot-18-lamdap-lekhakhit/botrian-005/ en: https://astra-ai.co/videos/sequences/arithmetic-sequence/difference-and-calculation-of-general-term/ es: https://astra-ai.co/es/videos/sucesiones/sucesion-aritmetica/termino-general-de-una-sucesion/ pl: https://astra-ai.co/pl/videos/ciagi/ciag-arytmetyczny/roznica-i-obliczanie-wyrazu-ogolnego/ fr: https://astra-ai.co/fr/videos/suites/suite-arithmetique/difference-et-calcul-du-terme-general/ de: https://astra-ai.co/de/videos/folgen/arithmetische-folge/differenz-und-berechnung-des-allgemeinen-glieds/ sl: https://astra-ai.co/sl/videos/zaporedja/aritmeticno-zaporedje/razlika-in-splosni-clen/ hr: https://astra-ai.co/hr/videos/nizovi/aritmeticki-niz/razlika-i-opci-clan/ fi: https://astra-ai.co/fi/videos/jonot/aritmeettinen-jono/erotus-ja-yleinen-termi/ pt: https://astra-ai.co/pt/videos/sequencias/sequencia-aritmetica/a-diferenca-e-o-termo-geral/ nl: https://astra-ai.co/nl/videos/reeksen/rekenkundige-reeks/het-verschil-en-de-algemene-term/ da: https://astra-ai.co/da/videos/foelger/aritmetisk-foelge/differens-og-beregning-af-det-generelle-led/ el: https://astra-ai.co/el/videos/akolouthies/arithmitiki-akolouthia/difference-and-calculation-of-the-general-term/ cs: https://astra-ai.co/cs/videos/sekvence/aritmeticka-posloupnost/rozdil-a-obecny-pojem/ no: https://astra-ai.co/no/videos/folger/aritmetisk-f-lge/differanse-og-beregning-av-det-generelle-leddet/ bg: https://astra-ai.co/bg/videos/posledovatelnosti/aritmetichna-posledovatelnost/razlikata-i-obshtiyat-termin/ id: https://astra-ai.co/id/videos/barisan/barisan-aritmetika/beda-dan-perhitungan-suku-umum/ hu: https://astra-ai.co/hu/videos/sorozatok/szamtani-sorozat/a-kulonbseg-es-az-altalanos-tag-szamitasa/ ro: https://astra-ai.co/ro/videos/siruri/progresie-aritmetica/lectie-video-2c1c6955/ sv: https://astra-ai.co/sv/videos/foljder/aritmetisk-talfoljd/differens-och-berakning-av-den-allmanna-termen/ uk: https://astra-ai.co/uk/videos/uk-topic-14/uk-chapter-18/uk-difference-and-calculation-of-general-term/ vi: https://astra-ai.co/vi/videos/vi-topic-14/vi-chapter-18/vi-difference-and-calculation-of-general-term/ sk: https://astra-ai.co/sk/videos/postupnosti/kapitola-013/lekcia-005/ lv: https://astra-ai.co/lv/videos/virknes/aritmetiska-virkne/stunda-2c1c6955/ sr: https://astra-ai.co/sr/videos/nizovi/poglavlje-013/lekcija-005/ tr: https://astra-ai.co/tr/videos/diziler/bolum-013/ders-005/ ms: https://astra-ai.co/ms/videos/jujukan/bab-18-matematik-asas/pelajaran-005/ my: https://astra-ai.co/my/videos/my-af25d53a/my-5a550787/2c1c6955/ lo: https://astra-ai.co/lo/videos/lo-af25d53a/lo-5a550787/2c1c6955/ km: https://astra-ai.co/km/videos/sequences/chapter-5a550787/lesson-2c1c6955/ zh-TW: https://astra-ai.co/zh-TW/videos/sequences/chapter-5a550787/lesson-2c1c6955/ zh-HK: https://astra-ai.co/zh-HK/videos/shuxue-kecheng-014/shuxue-zhang-018/shuxue-kecheng-005/ br: https://astra-ai.co/br/videos/sequencias/capitulo-matematica-018/video-matematica-005/ es-AR: https://astra-ai.co/es-AR/videos/sucesiones/capitulo-13/leccion-5/ en-PH: https://astra-ai.co/en-PH/videos/sequences/chapter-5a550787/lesson-2c1c6955/ en-IN: https://astra-ai.co/en-IN/videos/sequences/chapter-5a550787/difference-and-calculation-of-general-term/ es-PE: https://astra-ai.co/es-PE/videos/sucesiones/capitulo-video-5a550787/leccion-de-video-2c1c6955/ ms-BN: https://astra-ai.co/ms-BN/videos/jujukan/bab-matematik-13/pelajaran-2c1c6955/ pt-TL: https://astra-ai.co/pt-TL/videos/sucessoes/capitulo-de-video-013/licao-de-video-005/ mx: https://astra-ai.co/mx/videos/sucesiones/capitulo-13/leccion-5/ et: https://astra-ai.co/et/videos/jadad/aritmeetiline-jada/vahe-ja-uldliige/ es-CL: https://astra-ai.co/es-CL/videos/sucesiones/capitulo-13/termino-general-de-una-sucesion/ ko: https://astra-ai.co/ko/videos/sequences-ko/13/005/ ja: https://astra-ai.co/ja/videos/sequences-ja/ja-chapter-13/005/ --- # Difference and Calculation of the General Term ## INTRODUCTION In mathematics, the concept of a difference (or ratio) between terms and the calculation of the general term are crucial for understanding sequences and series, especially in the context of arithmetic and geometric sequences. The difference (or ratio) between terms of a sequence allows for the determination of the pattern of growth or decay of the sequence, which is fundamental for calculating the general term of the sequence. ## ARITHMETIC SEQUENCES In an arithmetic sequence, the difference between consecutive terms is constant and is called the **common difference (d)**. If a\_n is the n-th term of the sequence, then a\_n = a\_(n-1) + d. The general term (or n-th term) of an arithmetic sequence can be calculated using the formula: **a\_n = a\_1 + (n – 1)d** where a\_1 is the first term of the sequence. ## GEOMETRIC SEQUENCES In a geometric sequence, the ratio between consecutive terms is constant and is called the **common ratio (r)**. For a geometric sequence, a\_(n+1) / a\_n = r. The general term of a geometric sequence is calculated using the formula: **a\_n = a\_1 \* r^(n–1)** where a\_1 represents the first term of the sequence. ## EXAMPLE (FOR ARITHMETIC SEQUENCE): Suppose we have two terms of an arithmetic sequence: a\_3 = 9 and a\_5 = 13. To calculate the common difference 'd' and the first term a\_1, we use the steps described above. First, we calculate the common difference 'd': Since a\_5 = a\_3 + (5-3)d, 13 = 9 + 2d 13 - 9 = 2d 4 = 2d d = 4 / 2 = 2. Alternatively, using the formula mentioned in the original text (which calculates the average difference over the number of steps): d = (a\_5 – a\_3) / (5 – 3) = (13 – 9) / 2 = 4 / 2 = 2. Once we know the difference, we can calculate the first term a\_1 by considering one of the known terms, for example, a\_3. We use the formula for the n-th term: a\_n = a\_1 + (n – 1)d For a\_3: a\_3 = a\_1 + (3 – 1)d Substitute a\_3 = 9 and d = 2 and solve for a\_1: 9 = a\_1 + (2) \* 2 9 = a\_1 + 4 a\_1 = 9 – 4 a\_1 = 5 So, the first term a\_1 of the sequence is 5, and we have already calculated the common difference 'd' as 2. This example illustrates how, with minimal information, we can determine the key characteristics of an arithmetic sequence. ## CONCLUSION The common difference/ratio and the calculation of the general term are basic concepts in mathematics that provide tools for analyzing and understanding sequences. Their use extends beyond the boundaries of mathematics, enabling exploration, modeling, and problem-solving in many scientific branches.