document.write( "Question 1091756: Please help...
\n" ); document.write( "If {b n} is an arithmetic sequence, explain why {10^bn} must be geometric.
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Algebra.Com's Answer #706248 by ikleyn(52797)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "To prove that the sequence \"10%5Eb%5Bn%5D\" is a geometric progression, consider the ratio  \"%2810%5Eb%5Bn%2B1%5D%29%2F%2810%5Eb%5Bn%5D%29\".\r\n" );
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document.write( "Since  \"b%5Bn%2B1%5D\" is an AP, you have  \"b%5Bn%2B1%5D\" = \"b%5Bn%5D%2Bd\",  where d is the common difference of the AP, i.e. the constant term independent on \"n'.\r\n" );
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document.write( "Therefore,  \"%2810%5Eb%5Bn%2B1%5D%29%2F%2810%5Eb%5Bn%5D%29\" = \"%2810%5E%28b%5Bn%5D%2Bd%29%29%2F%2810%5Eb%5Bn%5D%29\" = \"10%5E%28b%5Bn%5D%2Bd-b%5Bn%5D%29\" = \"10%5Ed\".\r\n" );
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document.write( "Thus the ratio of two consecutive terms \"10%5Eb%5Bn%2B1%5D\"  and  \"10%5Eb%5Bn%5D\"  is the constant value  \"10%5Ed\".\r\n" );
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document.write( "By the definition, it means that the sequence  \"10%5Eb%5Bn%5D\"   is a geometric progression.\r\n" );
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\n" ); document.write( "There is a bunch of lessons on arithmetic progressions in this site:\r
\n" ); document.write( "\n" ); document.write( "    - Arithmetic progressions\r
\n" ); document.write( "\n" ); document.write( "    - The proofs of the formulas for arithmetic progressions \r
\n" ); document.write( "\n" ); document.write( "    - Problems on arithmetic progressions \r
\n" ); document.write( "\n" ); document.write( "    - Word problems on arithmetic progressions\r
\n" ); document.write( "\n" ); document.write( "    - Mathematical induction and arithmetic progressions\r
\n" ); document.write( "\n" ); document.write( "    - One characteristic property of arithmetic progressions\r
\n" ); document.write( "\n" ); document.write( "    - Solved problems on arithmetic progressions \r
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\n" ); document.write( "\n" ); document.write( "There is a bunch of lessons on geometric progressions in this site\r
\n" ); document.write( "\n" ); document.write( "    - Geometric progressions\r
\n" ); document.write( "\n" ); document.write( "    - The proofs of the formulas for geometric progressions \r
\n" ); document.write( "\n" ); document.write( "    - Problems on geometric progressions\r
\n" ); document.write( "\n" ); document.write( "    - Word problems on geometric progressions\r
\n" ); document.write( "\n" ); document.write( "    - One characteristic property of geometric progressions\r
\n" ); document.write( "\n" ); document.write( "    - Solved problems on geometric progressions\r
\n" ); document.write( "\n" ); document.write( "    - Fresh, sweet and crispy problem on arithmetic and geometric progressions\r
\n" ); document.write( "\n" ); document.write( "    - Mathematical induction and geometric progressions\r
\n" ); document.write( "\n" ); document.write( "    - Mathematical induction for sequences other than arithmetic or geometric\r
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\n" ); document.write( "\n" ); document.write( "Also,  you have this free of charge online textbook in ALGEBRA-II in this site\r
\n" ); document.write( "\n" ); document.write( "    - ALGEBRA-II - YOUR ONLINE TEXTBOOK.\r
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\n" ); document.write( "\n" ); document.write( "The referred lessons are the parts of this online textbook under the topics \"Arithmetic progressions\" and \"Geometric progressions\".\r
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\n" ); document.write( "\n" ); document.write( "Save the link to this textbook\r
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\n" ); document.write( "\n" ); document.write( "https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson\r
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\n" ); document.write( "\n" ); document.write( "into your archive and use when it is needed.\r
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