{"id":90224,"date":"2025-06-01T10:23:25","date_gmt":"2025-06-01T10:23:25","guid":{"rendered":"https:\/\/exam.pscnotes.com\/mcq\/?p=90224"},"modified":"2025-06-01T10:23:25","modified_gmt":"2025-06-01T10:23:25","slug":"on-simplification-the-product-x-yx%c2%b2-y%c2%b2x%e2%81%b4-y%e2%81%b4x%e2%81%b8-y%e2%81%b8-h","status":"publish","type":"post","link":"https:\/\/exam.pscnotes.com\/mcq\/on-simplification-the-product-x-yx%c2%b2-y%c2%b2x%e2%81%b4-y%e2%81%b4x%e2%81%b8-y%e2%81%b8-h\/","title":{"rendered":"On simplification the product (x + y)(x\u00b2 + y\u00b2)(x\u2074 + y\u2074)(x\u2078 + y\u2078) … h"},"content":{"rendered":"

On simplification the product (x + y)(x\u00b2 + y\u00b2)(x\u2074 + y\u2074)(x\u2078 + y\u2078) … how many such terms are there which will have only single x and rest ys?<\/p>\n

[amp_mcq option1=”21″ option2=”10″ option3=”20″ option4=”1″ correct=”option4″]<\/p>\n

\n
\n
This question was previously asked in<\/div>\n
UPSC CAPF – 2018<\/div>\n<\/div>\n
Download PDF<\/a>Attempt Online<\/a><\/div>\n<\/div>\n
\nThe product is (x + y)(x\u00b2 + y\u00b2)(x\u2074 + y\u2074)(x\u2078 + y\u2078)… Let’s consider a product of n terms: P_n = (x^2^0 + y^2^0)(x^2^1 + y^2^1)…(x^2^(n-1) + y^2^(n-1)). When expanding this product, each term is formed by choosing either the first term (x raised to a power of 2) or the second term (y raised to the same power of 2) from each bracket. For example, from the i-th bracket (x^2^i + y^2^i), we pick either x^2^i or y^2^i. A term in the expansion is the product of one chosen term from each of the n brackets. We are looking for terms with a “single x”, which means the power of x in the term must be 1 (x\u00b9). Let’s say we pick x^2^i from one bracket and y^2^j from all other brackets (j \u2260 i). The resulting term is (x^2^i) * (product of y^2^j for all j \u2260 i). The power of x in this term is 2^i. For this power to be 1 (x\u00b9), we must have 2^i = 1. This occurs only when i = 0. So, the only way to get a term with x\u00b9 is to choose x^2^0 = x from the first bracket (x+y) and y^2^j from all subsequent brackets (j = 1, 2, …, n-1). The resulting term is x * y\u00b2 * y\u2074 * y\u2078 * … * y^(2^(n-1)) = x * y^(2\u00b9 + 2\u00b2 + … + 2^(n-1)). The power of y is the sum of a geometric series: 2(2^(n-1) – 1)\/(2-1) = 2^n – 2. The term is x\u00b9y^(2^n – 2). This is the only term in the expansion that has x raised to the power of 1. Any other combination will result in x raised to a power other than 1 (e.g., choosing x from multiple brackets, or choosing x^2^i where i > 0 results in a power 2^i > 1). Thus, there is only exactly one such term. The number of such terms is 1.
\n<\/section>\n
\nIn the expansion of a product of factors of the form (x^a + y^a), terms are formed by selecting one element from each factor and multiplying them. To find terms with a specific power of x, analyze the possible combinations of selections from each factor.
\n<\/section>\n
\nThis type of product (x+y)(x\u00b2+y\u00b2)(x\u2074+y\u2074)… is related to the identity (x-y)(x+y) = x\u00b2-y\u00b2, (x\u00b2-y\u00b2)(x\u00b2+y\u00b2) = x\u2074-y\u2074, and generally (x^2^n – y^2^n)(x^2^n + y^2^n) = x^2^(n+1) – y^2^(n+1). If the product were multiplied by (x-y), it would telescope nicely to x^(2^n) – y^(2^n). However, the question asks about the terms in the expansion of the product itself.
\n<\/section>\n","protected":false},"excerpt":{"rendered":"

On simplification the product (x + y)(x\u00b2 + y\u00b2)(x\u2074 + y\u2074)(x\u2078 + y\u2078) … how many such terms are there which will have only single x and rest ys? [amp_mcq option1=”21″ option2=”10″ option3=”20″ option4=”1″ correct=”option4″] This question was previously asked in UPSC CAPF – 2018 Download PDFAttempt Online The product is (x + y)(x\u00b2 + … <\/p>\n

Detailed SolutionOn simplification the product (x + y)(x\u00b2 + y\u00b2)(x\u2074 + y\u2074)(x\u2078 + y\u2078) … h<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1085],"tags":[1114,1102],"class_list":["post-90224","post","type-post","status-publish","format-standard","hentry","category-upsc-capf","tag-1114","tag-quantitative-aptitude-and-reasoning","no-featured-image-padding"],"yoast_head":"\nOn simplification the product (x + y)(x\u00b2 + y\u00b2)(x\u2074 + y\u2074)(x\u2078 + y\u2078) ... h<\/title>\n<meta name=\"description\" content=\"The product is (x + y)(x\u00b2 + y\u00b2)(x\u2074 + y\u2074)(x\u2078 + y\u2078)... Let's consider a product of n terms: P_n = (x^2^0 + y^2^0)(x^2^1 + y^2^1)...(x^2^(n-1) + y^2^(n-1)). When expanding this product, each term is formed by choosing either the first term (x raised to a power of 2) or the second term (y raised to the same power of 2) from each bracket. For example, from the i-th bracket (x^2^i + y^2^i), we pick either x^2^i or y^2^i. A term in the expansion is the product of one chosen term from each of the n brackets. We are looking for terms with a "single x", which means the power of x in the term must be 1 (x\u00b9). Let's say we pick x^2^i from one bracket and y^2^j from all other brackets (j \u2260 i). The resulting term is (x^2^i) * (product of y^2^j for all j \u2260 i). The power of x in this term is 2^i. For this power to be 1 (x\u00b9), we must have 2^i = 1. This occurs only when i = 0. So, the only way to get a term with x\u00b9 is to choose x^2^0 = x from the first bracket (x+y) and y^2^j from all subsequent brackets (j = 1, 2, ..., n-1). The resulting term is x * y\u00b2 * y\u2074 * y\u2078 * ... * y^(2^(n-1)) = x * y^(2\u00b9 + 2\u00b2 + ... + 2^(n-1)). The power of y is the sum of a geometric series: 2(2^(n-1) - 1)\/(2-1) = 2^n - 2. The term is x\u00b9y^(2^n - 2). This is the only term in the expansion that has x raised to the power of 1. Any other combination will result in x raised to a power other than 1 (e.g., choosing x from multiple brackets, or choosing x^2^i where i > 0 results in a power 2^i > 1). Thus, there is only exactly one such term. The number of such terms is 1. In the expansion of a product of factors of the form (x^a + y^a), terms are formed by selecting one element from each factor and multiplying them. To find terms with a specific power of x, analyze the possible combinations of selections from each factor.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/exam.pscnotes.com\/mcq\/on-simplification-the-product-x-yx\u00b2-y\u00b2x\u2074-y\u2074x\u2078-y\u2078-h\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"On simplification the product (x + y)(x\u00b2 + y\u00b2)(x\u2074 + y\u2074)(x\u2078 + y\u2078) ... h\" \/>\n<meta property=\"og:description\" content=\"The product is (x + y)(x\u00b2 + y\u00b2)(x\u2074 + y\u2074)(x\u2078 + y\u2078)... Let's consider a product of n terms: P_n = (x^2^0 + y^2^0)(x^2^1 + y^2^1)...(x^2^(n-1) + y^2^(n-1)). When expanding this product, each term is formed by choosing either the first term (x raised to a power of 2) or the second term (y raised to the same power of 2) from each bracket. For example, from the i-th bracket (x^2^i + y^2^i), we pick either x^2^i or y^2^i. A term in the expansion is the product of one chosen term from each of the n brackets. We are looking for terms with a "single x", which means the power of x in the term must be 1 (x\u00b9). Let's say we pick x^2^i from one bracket and y^2^j from all other brackets (j \u2260 i). The resulting term is (x^2^i) * (product of y^2^j for all j \u2260 i). The power of x in this term is 2^i. For this power to be 1 (x\u00b9), we must have 2^i = 1. This occurs only when i = 0. So, the only way to get a term with x\u00b9 is to choose x^2^0 = x from the first bracket (x+y) and y^2^j from all subsequent brackets (j = 1, 2, ..., n-1). The resulting term is x * y\u00b2 * y\u2074 * y\u2078 * ... * y^(2^(n-1)) = x * y^(2\u00b9 + 2\u00b2 + ... + 2^(n-1)). The power of y is the sum of a geometric series: 2(2^(n-1) - 1)\/(2-1) = 2^n - 2. The term is x\u00b9y^(2^n - 2). This is the only term in the expansion that has x raised to the power of 1. Any other combination will result in x raised to a power other than 1 (e.g., choosing x from multiple brackets, or choosing x^2^i where i > 0 results in a power 2^i > 1). Thus, there is only exactly one such term. The number of such terms is 1. In the expansion of a product of factors of the form (x^a + y^a), terms are formed by selecting one element from each factor and multiplying them. To find terms with a specific power of x, analyze the possible combinations of selections from each factor.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/exam.pscnotes.com\/mcq\/on-simplification-the-product-x-yx\u00b2-y\u00b2x\u2074-y\u2074x\u2078-y\u2078-h\/\" \/>\n<meta property=\"og:site_name\" content=\"MCQ and Quiz for Exams\" \/>\n<meta property=\"article:published_time\" content=\"2025-06-01T10:23:25+00:00\" \/>\n<meta name=\"author\" content=\"rawan239\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"rawan239\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"2 minutes\" \/>\n<!-- \/ Yoast SEO Premium plugin. -->","yoast_head_json":{"title":"On simplification the product (x + y)(x\u00b2 + y\u00b2)(x\u2074 + y\u2074)(x\u2078 + y\u2078) ... h","description":"The product is (x + y)(x\u00b2 + y\u00b2)(x\u2074 + y\u2074)(x\u2078 + y\u2078)... Let's consider a product of n terms: P_n = (x^2^0 + y^2^0)(x^2^1 + y^2^1)...(x^2^(n-1) + y^2^(n-1)). When expanding this product, each term is formed by choosing either the first term (x raised to a power of 2) or the second term (y raised to the same power of 2) from each bracket. For example, from the i-th bracket (x^2^i + y^2^i), we pick either x^2^i or y^2^i. A term in the expansion is the product of one chosen term from each of the n brackets. We are looking for terms with a \"single x\", which means the power of x in the term must be 1 (x\u00b9). Let's say we pick x^2^i from one bracket and y^2^j from all other brackets (j \u2260 i). The resulting term is (x^2^i) * (product of y^2^j for all j \u2260 i). The power of x in this term is 2^i. For this power to be 1 (x\u00b9), we must have 2^i = 1. This occurs only when i = 0. So, the only way to get a term with x\u00b9 is to choose x^2^0 = x from the first bracket (x+y) and y^2^j from all subsequent brackets (j = 1, 2, ..., n-1). The resulting term is x * y\u00b2 * y\u2074 * y\u2078 * ... * y^(2^(n-1)) = x * y^(2\u00b9 + 2\u00b2 + ... + 2^(n-1)). The power of y is the sum of a geometric series: 2(2^(n-1) - 1)\/(2-1) = 2^n - 2. The term is x\u00b9y^(2^n - 2). This is the only term in the expansion that has x raised to the power of 1. Any other combination will result in x raised to a power other than 1 (e.g., choosing x from multiple brackets, or choosing x^2^i where i > 0 results in a power 2^i > 1). Thus, there is only exactly one such term. The number of such terms is 1. In the expansion of a product of factors of the form (x^a + y^a), terms are formed by selecting one element from each factor and multiplying them. To find terms with a specific power of x, analyze the possible combinations of selections from each factor.","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/exam.pscnotes.com\/mcq\/on-simplification-the-product-x-yx\u00b2-y\u00b2x\u2074-y\u2074x\u2078-y\u2078-h\/","og_locale":"en_US","og_type":"article","og_title":"On simplification the product (x + y)(x\u00b2 + y\u00b2)(x\u2074 + y\u2074)(x\u2078 + y\u2078) ... h","og_description":"The product is (x + y)(x\u00b2 + y\u00b2)(x\u2074 + y\u2074)(x\u2078 + y\u2078)... Let's consider a product of n terms: P_n = (x^2^0 + y^2^0)(x^2^1 + y^2^1)...(x^2^(n-1) + y^2^(n-1)). When expanding this product, each term is formed by choosing either the first term (x raised to a power of 2) or the second term (y raised to the same power of 2) from each bracket. For example, from the i-th bracket (x^2^i + y^2^i), we pick either x^2^i or y^2^i. A term in the expansion is the product of one chosen term from each of the n brackets. We are looking for terms with a \"single x\", which means the power of x in the term must be 1 (x\u00b9). Let's say we pick x^2^i from one bracket and y^2^j from all other brackets (j \u2260 i). The resulting term is (x^2^i) * (product of y^2^j for all j \u2260 i). The power of x in this term is 2^i. For this power to be 1 (x\u00b9), we must have 2^i = 1. This occurs only when i = 0. So, the only way to get a term with x\u00b9 is to choose x^2^0 = x from the first bracket (x+y) and y^2^j from all subsequent brackets (j = 1, 2, ..., n-1). The resulting term is x * y\u00b2 * y\u2074 * y\u2078 * ... * y^(2^(n-1)) = x * y^(2\u00b9 + 2\u00b2 + ... + 2^(n-1)). The power of y is the sum of a geometric series: 2(2^(n-1) - 1)\/(2-1) = 2^n - 2. The term is x\u00b9y^(2^n - 2). This is the only term in the expansion that has x raised to the power of 1. Any other combination will result in x raised to a power other than 1 (e.g., choosing x from multiple brackets, or choosing x^2^i where i > 0 results in a power 2^i > 1). Thus, there is only exactly one such term. The number of such terms is 1. In the expansion of a product of factors of the form (x^a + y^a), terms are formed by selecting one element from each factor and multiplying them. To find terms with a specific power of x, analyze the possible combinations of selections from each factor.","og_url":"https:\/\/exam.pscnotes.com\/mcq\/on-simplification-the-product-x-yx\u00b2-y\u00b2x\u2074-y\u2074x\u2078-y\u2078-h\/","og_site_name":"MCQ and Quiz for Exams","article_published_time":"2025-06-01T10:23:25+00:00","author":"rawan239","twitter_card":"summary_large_image","twitter_misc":{"Written by":"rawan239","Est. reading time":"2 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"WebPage","@id":"https:\/\/exam.pscnotes.com\/mcq\/on-simplification-the-product-x-yx%c2%b2-y%c2%b2x%e2%81%b4-y%e2%81%b4x%e2%81%b8-y%e2%81%b8-h\/","url":"https:\/\/exam.pscnotes.com\/mcq\/on-simplification-the-product-x-yx%c2%b2-y%c2%b2x%e2%81%b4-y%e2%81%b4x%e2%81%b8-y%e2%81%b8-h\/","name":"On simplification the product (x + y)(x\u00b2 + y\u00b2)(x\u2074 + y\u2074)(x\u2078 + y\u2078) ... h","isPartOf":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#website"},"datePublished":"2025-06-01T10:23:25+00:00","dateModified":"2025-06-01T10:23:25+00:00","author":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#\/schema\/person\/5807dafeb27d2ec82344d6cbd6c3d209"},"description":"The product is (x + y)(x\u00b2 + y\u00b2)(x\u2074 + y\u2074)(x\u2078 + y\u2078)... Let's consider a product of n terms: P_n = (x^2^0 + y^2^0)(x^2^1 + y^2^1)...(x^2^(n-1) + y^2^(n-1)). When expanding this product, each term is formed by choosing either the first term (x raised to a power of 2) or the second term (y raised to the same power of 2) from each bracket. For example, from the i-th bracket (x^2^i + y^2^i), we pick either x^2^i or y^2^i. A term in the expansion is the product of one chosen term from each of the n brackets. We are looking for terms with a \"single x\", which means the power of x in the term must be 1 (x\u00b9). Let's say we pick x^2^i from one bracket and y^2^j from all other brackets (j \u2260 i). The resulting term is (x^2^i) * (product of y^2^j for all j \u2260 i). The power of x in this term is 2^i. For this power to be 1 (x\u00b9), we must have 2^i = 1. This occurs only when i = 0. So, the only way to get a term with x\u00b9 is to choose x^2^0 = x from the first bracket (x+y) and y^2^j from all subsequent brackets (j = 1, 2, ..., n-1). The resulting term is x * y\u00b2 * y\u2074 * y\u2078 * ... * y^(2^(n-1)) = x * y^(2\u00b9 + 2\u00b2 + ... + 2^(n-1)). The power of y is the sum of a geometric series: 2(2^(n-1) - 1)\/(2-1) = 2^n - 2. The term is x\u00b9y^(2^n - 2). This is the only term in the expansion that has x raised to the power of 1. Any other combination will result in x raised to a power other than 1 (e.g., choosing x from multiple brackets, or choosing x^2^i where i > 0 results in a power 2^i > 1). Thus, there is only exactly one such term. The number of such terms is 1. In the expansion of a product of factors of the form (x^a + y^a), terms are formed by selecting one element from each factor and multiplying them. 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