{"id":294233,"date":"2022-09-10T19:54:39","date_gmt":"2022-09-10T14:24:39","guid":{"rendered":"https:\/\/infinitylearn.com\/surge\/question\/let-a10-and-a1a2a3an-be-real-numbers-such-that-aiai%e2%88%9211-for-all-i-then-the-a-m-of-the-numbers-a1a2a3an-has-the\/"},"modified":"2022-09-10T19:54:39","modified_gmt":"2022-09-10T14:24:39","slug":"let-a10-and-a1a2a3an-be-real-numbers-such-that-aiai%e2%88%9211-for-all-i-then-the-a-m-of-the-numbers-a1a2a3an-has-the","status":"publish","type":"questions","link":"https:\/\/infinitylearn.com\/surge\/question\/mathematics\/leta10anda1a2a3anbe-real-numbers-such-thataiai1\/","title":{"rendered":"Let\u00a0a1=0\u00a0and\u00a0a1,a2,a3,\u2026,an\u00a0be real numbers such that\u00a0ai=ai\u22121+1\u00a0for all\u00a0\u00a0i, then the A.M. of the numbers\u00a0a1,a2,a3,\u2026,an\u00a0has the value\u00a0A\u00a0where"},"content":{"rendered":"","protected":false},"author":1,"template":"","meta":{"_yoast_wpseo_focuskw":"","_yoast_wpseo_title":"","_yoast_wpseo_metadesc":"Let\u00a0a1=0\u00a0and\u00a0a1,a2,a3,\u2026,an\u00a0be real numbers such that\u00a0ai=ai\u22121+1\u00a0for all\u00a0\u00a0i, then the A.M. of the numbers\u00a0a1,a2,a3,\u2026,an\u00a0has the value\u00a0A\u00a0where","custom_permalink":"question\/mathematics\/leta10anda1a2a3anbe-real-numbers-such-thataiai1\/"},"categories":[],"acf":{"question":"<p>Let&nbsp;<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>a<\/mi><mn>1<\/mn><\/msub><mo>=<\/mo><mn>0<\/mn><mtext>&nbsp;<\/mtext><\/math>and&nbsp;<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>a<\/mi><mn>1<\/mn><\/msub><mo>,<\/mo><msub><mi>a<\/mi><mn>2<\/mn><\/msub><mo>,<\/mo><msub><mi>a<\/mi><mn>3<\/mn><\/msub><mo>,<\/mo><mo>\u2026<\/mo><mo>,<\/mo><msub><mi>a<\/mi><mi>n<\/mi><\/msub><\/math>&nbsp;be real numbers such that&nbsp;<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mfenced separators=\"|\" close=\"|\" open=\"|\"><msub><mi>a<\/mi><mi>i<\/mi><\/msub><\/mfenced><mo>=<\/mo><mfenced separators=\"|\" close=\"|\" open=\"|\"><mrow><msub><mi>a<\/mi><mrow><mi>i<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><mo>+<\/mo><mn>1<\/mn><\/mrow><\/mfenced><\/math>&nbsp;for all&nbsp;<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mtext>&nbsp;<\/mtext><mi>i<\/mi><\/math>, then the A.M. of the numbers&nbsp;<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>a<\/mi><mn>1<\/mn><\/msub><mo>,<\/mo><msub><mi>a<\/mi><mn>2<\/mn><\/msub><mo>,<\/mo><msub><mi>a<\/mi><mn>3<\/mn><\/msub><mo>,<\/mo><mo>\u2026<\/mo><mo>,<\/mo><msub><mi>a<\/mi><mi>n<\/mi><\/msub><\/math>&nbsp;has the value&nbsp;<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>A<\/mi><\/math>&nbsp;where<\/p>","options":[{"option":"<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>A<\/mi><mo>&lt;<\/mo><mo>\u2212<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><\/math><\/p>","correct":false},{"option":"<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>A<\/mi><mo>&lt;<\/mo><mo>\u2212<\/mo><mn>1<\/mn><\/math><\/p>","correct":false},{"option":"<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>A<\/mi><mo>\u2265<\/mo><mo>\u2212<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><\/math><\/p>","correct":true},{"option":"<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>A<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><\/math><\/p>","correct":false}],"solution":"<p>We have,<\/p><p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mfenced separators=\"|\" close=\"|\" open=\"|\"><msub><mi>a<\/mi><mi>i<\/mi><\/msub><\/mfenced><mo>=<\/mo><mfenced separators=\"|\" close=\"|\" open=\"|\"><mrow><msub><mi>a<\/mi><mrow><mi>i<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><mo>+<\/mo><mn>1<\/mn><\/mrow><\/mfenced><mtext>&nbsp;<\/mtext><\/math>&nbsp;for all&nbsp;<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mo>&nbsp;<\/mo><mi>i<\/mi><mo>=<\/mo><mn>2<\/mn><mo>,<\/mo><mn>3<\/mn><mo>,<\/mo><mo>\u2026<\/mo><mi>n<\/mi><mo>+<\/mo><mn>1<\/mn><\/math>&nbsp;&nbsp;<\/p><p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mtable columnalign=\"left\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\"><mtr><mtd><mo stretchy=\"false\">\u21d2<\/mo><mspace width=\"1em\">&nbsp;<\/mspace><msubsup><mi>a<\/mi><mi>i<\/mi><mn>2<\/mn><\/msubsup><mo>=<\/mo><msup><mfenced separators=\"|\"><mrow><msub><mi>a<\/mi><mrow><mi>i<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><mo>+<\/mo><mn>1<\/mn><\/mrow><\/mfenced><mn>2<\/mn><\/msup><\/mtd><\/mtr><mtr><mtd><mo stretchy=\"false\">\u21d2<\/mo><mspace width=\"1em\">&nbsp;<\/mspace><msubsup><mi>a<\/mi><mi>i<\/mi><mn>2<\/mn><\/msubsup><mo>\u2212<\/mo><msubsup><mi>a<\/mi><mrow><mi>i<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><mn>2<\/mn><\/msubsup><mo>=<\/mo><mn>2<\/mn><msub><mi>a<\/mi><mrow><mi>i<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><mo>+<\/mo><mn>1<\/mn><\/mtd><\/mtr><mtr><mtd><mo stretchy=\"false\">\u21d2<\/mo><mspace width=\"1em\">&nbsp;<\/mspace><munderover><mo>\u2211<\/mo><mrow><mi>i<\/mi><mo>=<\/mo><mn>2<\/mn><\/mrow><mrow><mi>n<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><\/munderover><mo>\u200a<\/mo><msubsup><mi>a<\/mi><mi>i<\/mi><mn>2<\/mn><\/msubsup><mo>\u2212<\/mo><msubsup><mi>a<\/mi><mrow><mi>i<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><mn>2<\/mn><\/msubsup><mo>=<\/mo><munderover><mo>\u2211<\/mo><mrow><mi>i<\/mi><mo>=<\/mo><mn>2<\/mn><\/mrow><mrow><mi>n<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><\/munderover><mo>\u200a<\/mo><mfenced separators=\"|\"><mrow><mn>2<\/mn><msub><mi>a<\/mi><mrow><mi>i<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><mo>+<\/mo><mn>1<\/mn><\/mrow><\/mfenced><\/mtd><\/mtr><mtr><mtd><mo stretchy=\"false\">\u21d2<\/mo><msubsup><mi>a<\/mi><mrow><mi>n<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><mn>2<\/mn><\/msubsup><mo>\u2212<\/mo><msubsup><mi>a<\/mi><mn>1<\/mn><mn>2<\/mn><\/msubsup><mo>=<\/mo><mn>2<\/mn><mfenced separators=\"|\"><mrow><msub><mi>a<\/mi><mn>1<\/mn><\/msub><mo>+<\/mo><msub><mi>a<\/mi><mn>2<\/mn><\/msub><mo>+<\/mo><mo>\u2026<\/mo><mo>+<\/mo><msub><mi>a<\/mi><mi>n<\/mi><\/msub><\/mrow><\/mfenced><mo>+<\/mo><mi>n<\/mi><\/mtd><\/mtr><mtr><mtd><mo stretchy=\"false\">\u21d2<\/mo><mspace width=\"1em\">&nbsp;<\/mspace><msubsup><mi>a<\/mi><mrow><mi>n<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><mn>2<\/mn><\/msubsup><mo>=<\/mo><mn>2<\/mn><mi>n<\/mi><mi>A<\/mi><mo>+<\/mo><mi>n<\/mi><mspace width=\"1em\">&nbsp;<\/mspace><mfenced separators=\"|\" close=\"]\" open=\"[\"><mrow><mo>\u2235<\/mo><mi>A<\/mi><mo>=<\/mo><mfrac><mrow><msub><mi>a<\/mi><mn>1<\/mn><\/msub><mo>+<\/mo><msub><mi>a<\/mi><mn>2<\/mn><\/msub><mo>+<\/mo><mo>\u2026<\/mo><mo>+<\/mo><msub><mi>a<\/mi><mi>n<\/mi><\/msub><\/mrow><mi>n<\/mi><\/mfrac><mtext>&nbsp;and&nbsp;<\/mtext><msub><mi>a<\/mi><mn>1<\/mn><\/msub><mo>=<\/mo><mn>0<\/mn><\/mrow><\/mfenced><\/mtd><\/mtr><mtr><mtd><mo stretchy=\"false\">\u21d2<\/mo><mspace width=\"1em\">&nbsp;<\/mspace><mi>A<\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mrow><mn>2<\/mn><mi>n<\/mi><\/mrow><\/mfrac><mfenced separators=\"|\"><mrow><msubsup><mi>a<\/mi><mrow><mi>n<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><mn>2<\/mn><\/msubsup><mo>\u2212<\/mo><mi>n<\/mi><\/mrow><\/mfenced><mo stretchy=\"false\">\u21d2<\/mo><mi>A<\/mi><mo>=<\/mo><mfrac><msubsup><mi>a<\/mi><mrow><mi>n<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><mn>2<\/mn><\/msubsup><mrow><mn>2<\/mn><mi>n<\/mi><\/mrow><\/mfrac><mo>\u2212<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mo stretchy=\"false\">\u21d2<\/mo><mi>A<\/mi><mo>\u2265<\/mo><mo>\u2212<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><\/mtd><\/mtr><\/mtable><\/math><\/p>","subject":""},"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v17.9 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Let\u00a0a1=0\u00a0and\u00a0a1,a2,a3,\u2026,an\u00a0be real numbers such that\u00a0ai=ai\u22121+1\u00a0for all\u00a0\u00a0i, then the A.M. of the numbers\u00a0a1,a2,a3,\u2026,an\u00a0has the value\u00a0A\u00a0where - Infinity Learn by Sri Chaitanya<\/title>\n<meta name=\"description\" content=\"Let\u00a0a1=0\u00a0and\u00a0a1,a2,a3,\u2026,an\u00a0be real numbers such that\u00a0ai=ai\u22121+1\u00a0for all\u00a0\u00a0i, then the A.M. of the numbers\u00a0a1,a2,a3,\u2026,an\u00a0has the value\u00a0A\u00a0where\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" 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value\u00a0A\u00a0where","breadcrumb":{"@id":"https:\/\/infinitylearn.com\/surge\/question\/mathematics\/leta10anda1a2a3anbe-real-numbers-such-thataiai1\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/infinitylearn.com\/surge\/question\/mathematics\/leta10anda1a2a3anbe-real-numbers-such-thataiai1\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/infinitylearn.com\/surge\/question\/mathematics\/leta10anda1a2a3anbe-real-numbers-such-thataiai1\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/infinitylearn.com\/surge\/"},{"@type":"ListItem","position":2,"name":"Let\u00a0a1=0\u00a0and\u00a0a1,a2,a3,\u2026,an\u00a0be real numbers such that\u00a0ai=ai\u22121+1\u00a0for all\u00a0\u00a0i, then the A.M. of the numbers\u00a0a1,a2,a3,\u2026,an\u00a0has the 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