Common Ratio
The constant factor between consecutive terms of a geometric sequence is called the common ratio.
Example:
Given the geometric sequence
%22%3E%3Cg%20data-mml-node%3D%22math%22%3E%3Cg%20data-mml-node%3D%22mn%22%3E%3Cpath%20data-c%3D%2232%22%20d%3D%22M109%20429Q82%20429%2066%20447T50%20491Q50%20562%20103%20614T235%20666Q326%20666%20387%20610T449%20465Q449%20422%20429%20383T381%20315T301%20241Q265%20210%20201%20149L142%2093L218%2092Q375%2092%20385%2097Q392%2099%20409%20186V189H449V186Q448%20183%20436%2095T421%203V0H50V19V31Q50%2038%2056%2046T86%2081Q115%20113%20136%20137Q145%20147%20170%20174T204%20211T233%20244T261%20278T284%20308T305%20340T320%20369T333%20401T340%20431T343%20464Q343%20527%20309%20573T212%20619Q179%20619%20154%20602T119%20569T109%20550Q109%20549%20114%20549Q132%20549%20151%20535T170%20489Q170%20464%20154%20447T109%20429Z%22%3E%3C%2Fpath%3E%3C%2Fg%3E%3Cg%20data-mml-node%3D%22mo%22%20transform%3D%22translate(500%2C0)%22%3E%3Cpath%20data-c%3D%222C%22%20d%3D%22M78%2035T78%2060T94%20103T137%20121Q165%20121%20187%2096T210%208Q210%20-27%20201%20-60T180%20-117T154%20-158T130%20-185T117%20-194Q113%20-194%20104%20-185T95%20-172Q95%20-168%20106%20-156T131%20-126T157%20-76T173%20-3V9L172%208Q170%207%20167%206T161%203T152%201T140%200Q113%200%2096%2017Z%22%3E%3C%2Fpath%3E%3C%2Fg%3E%3Cg%20data-mml-node%3D%22mn%22%20transform%3D%22translate(944.7%2C0)%22%3E%3Cpath%20data-c%3D%2234%22%20d%3D%22M462%200Q444%203%20333%203Q217%203%20199%200H190V46H221Q241%2046%20248%2046T265%2048T279%2053T286%2061Q287%2063%20287%20115V165H28V211L179%20442Q332%20674%20334%20675Q336%20677%20355%20677H373L379%20671V211H471V165H379V114Q379%2073%20379%2066T385%2054Q393%2047%20442%2046H471V0H462ZM293%20211V545L74%20212L183%20211H293Z%22%3E%3C%2Fpath%3E%3C%2Fg%3E%3Cg%20data-mml-node%3D%22mo%22%20transform%3D%22translate(1444.7%2C0)%22%3E%3Cpath%20data-c%3D%222C%22%20d%3D%22M78%2035T78%2060T94%20103T137%20121Q165%20121%20187%2096T210%208Q210%20-27%20201%20-60T180%20-117T154%20-158T130%20-185T117%20-194Q113%20-194%20104%20-185T95%20-172Q95%20-168%20106%20-156T131%20-126T157%20-76T173%20-3V9L172%208Q170%207%20167%206T161%203T152%201T140%200Q113%200%2096%2017Z%22%3E%3C%2Fpath%3E%3C%2Fg%3E%3Cg%20data-mml-node%3D%22mn%22%20transform%3D%22translate(1889.3%2C0)%22%3E%3Cpath%20data-c%3D%2238%22%20d%3D%22M70%20417T70%20494T124%20618T248%20666Q319%20666%20374%20624T429%20515Q429%20485%20418%20459T392%20417T361%20389T335%20371T324%20363L338%20354Q352%20344%20366%20334T382%20323Q457%20264%20457%20174Q457%2095%20399%2037T249%20-22Q159%20-22%20101%2029T43%20155Q43%20263%20172%20335L154%20348Q133%20361%20127%20368Q70%20417%2070%20494ZM286%20386L292%20390Q298%20394%20301%20396T311%20403T323%20413T334%20425T345%20438T355%20454T364%20471T369%20491T371%20513Q371%20556%20342%20586T275%20624Q268%20625%20242%20625Q201%20625%20165%20599T128%20534Q128%20511%20141%20492T167%20463T217%20431Q224%20426%20228%20424L286%20386ZM250%2021Q308%2021%20350%2055T392%20137Q392%20154%20387%20169T375%20194T353%20216T330%20234T301%20253T274%20270Q260%20279%20244%20289T218%20306L210%20311Q204%20311%20181%20294T133%20239T107%20157Q107%2098%20150%2060T250%2021Z%22%3E%3C%2Fpath%3E%3C%2Fg%3E%3Cg%20data-mml-node%3D%22mo%22%20transform%3D%22translate(2389.3%2C0)%22%3E%3Cpath%20data-c%3D%222C%22%20d%3D%22M78%2035T78%2060T94%20103T137%20121Q165%20121%20187%2096T210%208Q210%20-27%20201%20-60T180%20-117T154%20-158T130%20-185T117%20-194Q113%20-194%20104%20-185T95%20-172Q95%20-168%20106%20-156T131%20-126T157%20-76T173%20-3V9L172%208Q170%207%20167%206T161%203T152%201T140%200Q113%200%2096%2017Z%22%3E%3C%2Fpath%3E%3C%2Fg%3E%3Cg%20data-mml-node%3D%22mn%22%20transform%3D%22translate(2834%2C0)%22%3E%3Cpath%20data-c%3D%2231%22%20d%3D%22M213%20578L200%20573Q186%20568%20160%20563T102%20556H83V602H102Q149%20604%20189%20617T245%20641T273%20663Q275%20666%20285%20666Q294%20666%20302%20660V361L303%2061Q310%2054%20315%2052T339%2048T401%2046H427V0H416Q395%203%20257%203Q121%203%20100%200H88V46H114Q136%2046%20152%2046T177%2047T193%2050T201%2052T207%2057T213%2061V578Z%22%3E%3C%2Fpath%3E%3Cpath%20data-c%3D%2236%22%20d%3D%22M42%20313Q42%20476%20123%20571T303%20666Q372%20666%20402%20630T432%20550Q432%20525%20418%20510T379%20495Q356%20495%20341%20509T326%20548Q326%20592%20373%20601Q351%20623%20311%20626Q240%20626%20194%20566Q147%20500%20147%20364L148%20360Q153%20366%20156%20373Q197%20433%20263%20433H267Q313%20433%20348%20414Q372%20400%20396%20374T435%20317Q456%20268%20456%20210V192Q456%20169%20451%20149Q440%2090%20387%2034T253%20-22Q225%20-22%20199%20-14T143%2016T92%2075T56%20172T42%20313ZM257%20397Q227%20397%20205%20380T171%20335T154%20278T148%20216Q148%20133%20160%2097T198%2039Q222%2021%20251%2021Q302%2021%20329%2059Q342%2077%20347%20104T352%20209Q352%20289%20347%20316T329%20361Q302%20397%20257%20397Z%22%20transform%3D%22translate(500%2C0)%22%3E%3C%2Fpath%3E%3C%2Fg%3E%3Cg%20data-mml-node%3D%22mo%22%20transform%3D%22translate(3834%2C0)%22%3E%3Cpath%20data-c%3D%222C%22%20d%3D%22M78%2035T78%2060T94%20103T137%20121Q165%20121%20187%2096T210%208Q210%20-27%20201%20-60T180%20-117T154%20-158T130%20-185T117%20-194Q113%20-194%20104%20-185T95%20-172Q95%20-168%20106%20-156T131%20-126T157%20-76T173%20-3V9L172%208Q170%207%20167%206T161%203T152%201T140%200Q113%200%2096%2017Z%22%3E%3C%2Fpath%3E%3C%2Fg%3E%3Cg%20data-mml-node%3D%22mn%22%20transform%3D%22translate(4278.7%2C0)%22%3E%3Cpath%20data-c%3D%222E%22%20d%3D%22M78%2060Q78%2084%2095%20102T138%20120Q162%20120%20180%20104T199%2061Q199%2036%20182%2018T139%200T96%2017T78%2060Z%22%3E%3C%2Fpath%3E%3Cpath%20data-c%3D%222E%22%20d%3D%22M78%2060Q78%2084%2095%20102T138%20120Q162%20120%20180%20104T199%2061Q199%2036%20182%2018T139%200T96%2017T78%2060Z%22%20transform%3D%22translate(278%2C0)%22%3E%3C%2Fpath%3E%3Cpath%20data-c%3D%222E%22%20d%3D%22M78%2060Q78%2084%2095%20102T138%20120Q162%20120%20180%20104T199%2061Q199%2036%20182%2018T139%200T96%2017T78%2060Z%22%20transform%3D%22translate(556%2C0)%22%3E%3C%2Fpath%3E%3C%2Fg%3E%3C%2Fg%3E%3C%2Fg%3E%3E%3C%2Fsvg%3E)
.
To find the common ratio , find the ratio between a term and the term preceding it.
%22%3E%3Cg%20data-mml-node%3D%22math%22%3E%3Cg%20data-mml-node%3D%22mrow%22%3E%3Cg%20data-mml-node%3D%22mi%22%3E%3Cpath%20data-c%3D%221D45F%22%20d%3D%22M21%20287Q22%20290%2023%20295T28%20317T38%20348T53%20381T73%20411T99%20433T132%20442Q161%20442%20183%20430T214%20408T225%20388Q227%20382%20228%20382T236%20389Q284%20441%20347%20441H350Q398%20441%20422%20400Q430%20381%20430%20363Q430%20333%20417%20315T391%20292T366%20288Q346%20288%20334%20299T322%20328Q322%20376%20378%20392Q356%20405%20342%20405Q286%20405%20239%20331Q229%20315%20224%20298T190%20165Q156%2025%20151%2016Q138%20-11%20108%20-11Q95%20-11%2087%20-5T76%207T74%2017Q74%2030%20114%20189T154%20366Q154%20405%20128%20405Q107%20405%2092%20377T68%20316T57%20280Q55%20278%2041%20278H27Q21%20284%2021%20287Z%22%3E%3C%2Fpath%3E%3C%2Fg%3E%3Cg%20data-mml-node%3D%22mo%22%20transform%3D%22translate(728.8%2C0)%22%3E%3Cpath%20data-c%3D%223D%22%20d%3D%22M56%20347Q56%20360%2070%20367H707Q722%20359%20722%20347Q722%20336%20708%20328L390%20327H72Q56%20332%2056%20347ZM56%20153Q56%20168%2072%20173H708Q722%20163%20722%20153Q722%20140%20707%20133H70Q56%20140%2056%20153Z%22%3E%3C%2Fpath%3E%3C%2Fg%3E%3Cg%20data-mml-node%3D%22mfrac%22%20transform%3D%22translate(1784.6%2C0)%22%3E%3Cg%20data-mml-node%3D%22mn%22%20transform%3D%22translate(220%2C394)%20scale(0.707)%22%3E%3Cpath%20data-c%3D%2234%22%20d%3D%22M462%200Q444%203%20333%203Q217%203%20199%200H190V46H221Q241%2046%20248%2046T265%2048T279%2053T286%2061Q287%2063%20287%20115V165H28V211L179%20442Q332%20674%20334%20675Q336%20677%20355%20677H373L379%20671V211H471V165H379V114Q379%2073%20379%2066T385%2054Q393%2047%20442%2046H471V0H462ZM293%20211V545L74%20212L183%20211H293Z%22%3E%3C%2Fpath%3E%3C%2Fg%3E%3Cg%20data-mml-node%3D%22mn%22%20transform%3D%22translate(220%2C-345)%20scale(0.707)%22%3E%3Cpath%20data-c%3D%2232%22%20d%3D%22M109%20429Q82%20429%2066%20447T50%20491Q50%20562%20103%20614T235%20666Q326%20666%20387%20610T449%20465Q449%20422%20429%20383T381%20315T301%20241Q265%20210%20201%20149L142%2093L218%2092Q375%2092%20385%2097Q392%2099%20409%20186V189H449V186Q448%20183%20436%2095T421%203V0H50V19V31Q50%2038%2056%2046T86%2081Q115%20113%20136%20137Q145%20147%20170%20174T204%20211T233%20244T261%20278T284%20308T305%20340T320%20369T333%20401T340%20431T343%20464Q343%20527%20309%20573T212%20619Q179%20619%20154%20602T119%20569T109%20550Q109%20549%20114%20549Q132%20549%20151%20535T170%20489Q170%20464%20154%20447T109%20429Z%22%3E%3C%2Fpath%3E%3C%2Fg%3E%3Crect%20width%3D%22553.6%22%20height%3D%2260%22%20x%3D%22120%22%20y%3D%22220%22%3E%3C%2Frect%3E%3C%2Fg%3E%3Cg%20data-mml-node%3D%22mo%22%20transform%3D%22translate(2855.9%2C0)%22%3E%3Cpath%20data-c%3D%223D%22%20d%3D%22M56%20347Q56%20360%2070%20367H707Q722%20359%20722%20347Q722%20336%20708%20328L390%20327H72Q56%20332%2056%20347ZM56%20153Q56%20168%2072%20173H708Q722%20163%20722%20153Q722%20140%20707%20133H70Q56%20140%2056%20153Z%22%3E%3C%2Fpath%3E%3C%2Fg%3E%3Cg%20data-mml-node%3D%22mn%22%20transform%3D%22translate(3911.7%2C0)%22%3E%3Cpath%20data-c%3D%2232%22%20d%3D%22M109%20429Q82%20429%2066%20447T50%20491Q50%20562%20103%20614T235%20666Q326%20666%20387%20610T449%20465Q449%20422%20429%20383T381%20315T301%20241Q265%20210%20201%20149L142%2093L218%2092Q375%2092%20385%2097Q392%2099%20409%20186V189H449V186Q448%20183%20436%2095T421%203V0H50V19V31Q50%2038%2056%2046T86%2081Q115%20113%20136%20137Q145%20147%20170%20174T204%20211T233%20244T261%20278T284%20308T305%20340T320%20369T333%20401T340%20431T343%20464Q343%20527%20309%20573T212%20619Q179%20619%20154%20602T119%20569T109%20550Q109%20549%20114%20549Q132%20549%20151%20535T170%20489Q170%20464%20154%20447T109%20429Z%22%3E%3C%2Fpath%3E%3C%2Fg%3E%3C%2Fg%3E%3C%2Fg%3E%3C%2Fg%3E%3E%3C%2Fsvg%3E)
%22%3E%3Cg%20data-mml-node%3D%22math%22%3E%3Cg%20data-mml-node%3D%22mn%22%3E%3Cpath%20data-c%3D%2232%22%20d%3D%22M109%20429Q82%20429%2066%20447T50%20491Q50%20562%20103%20614T235%20666Q326%20666%20387%20610T449%20465Q449%20422%20429%20383T381%20315T301%20241Q265%20210%20201%20149L142%2093L218%2092Q375%2092%20385%2097Q392%2099%20409%20186V189H449V186Q448%20183%20436%2095T421%203V0H50V19V31Q50%2038%2056%2046T86%2081Q115%20113%20136%20137Q145%20147%20170%20174T204%20211T233%20244T261%20278T284%20308T305%20340T320%20369T333%20401T340%20431T343%20464Q343%20527%20309%20573T212%20619Q179%20619%20154%20602T119%20569T109%20550Q109%20549%20114%20549Q132%20549%20151%20535T170%20489Q170%20464%20154%20447T109%20429Z%22%3E%3C%2Fpath%3E%3C%2Fg%3E%3C%2Fg%3E%3C%2Fg%3E%3E%3C%2Fsvg%3E)
is the
common ratio.
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