document.write( "Question 1184007: prove by mathematical induction:
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Algebra.Com's Answer #852303 by ikleyn(52832) $\"\"$ $\"About$

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document.write( "(a) The base of induction: n = 1.\r\n" );
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document.write( "    Then the sum is one single term  ,  which is 1.\r\n" );
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document.write( "    The formula (*) at n = 1 gives \r\n" );
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document.write( "         =  =  =  = 1,  \r\n" );
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document.write( "    so the base of induction is established.\r\n" );
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document.write( "(b)  The step of induction.\r\n" );
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document.write( "     We assume that for some integer k >= 1 this formula is valid\r\n" );
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document.write( "         1^5 + 2^5 + 3^5 + . . . + k^5 = .     (1)\r\n" );
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document.write( "     We want to prove that then the formula is valid for the next integer number k+1, too:\r\n" );
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document.write( "         1^5 + 2^5 + 3^5 + . . . + k^5 + (k+1)^5 = .     (2)\r\n" );
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document.write( "     At this point, the proof of the formula (2) is started.\r\n" );
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document.write( "     In the left side of (2), we replace the sum of the first k addends by the right side expression (1).\r\n" );
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document.write( "     Thus we want to prove\r\n" );
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document.write( "          +  = .     (3)\r\n" );
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document.write( "     Let's transform left side of (3). We factor it, taking the common factor  out of parentheses.\r\n" );
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document.write( "     Then left side of (3) takes the form\r\n" );
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document.write( "            = \r\n" );
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document.write( "         =  = \r\n" );
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document.write( "         = .    (4)\r\n" );
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document.write( "    \r\n" );
document.write( "     Now, I used an online calculator to factor an expression in the internal parentheses,\r\n" );
document.write( "     and the calculator produced this decomposition\r\n" );
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document.write( "          = .     (5)\r\n" );
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document.write( "     ( the link to the calculator is  https://www.pocketmath.net , the mode is \"Factor\" )  )\r\n" );
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document.write( "     This factorization can be continued this way\r\n" );
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document.write( "          =  = .    (6)\r\n" );
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document.write( "     Now, combining all pieces (4), (5) and (6) in one whole block, we have\r\n" );
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document.write( "          +  = .   (7)\r\n" );
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document.write( "     It is the same as (identical to) formula (3).  Thus formula (3) is proven.\r\n" );
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document.write( "(3)  Due to the principle of the mathematical induction, it means that formula \r\n" );
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document.write( "          = .     \r\n" );
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document.write( "     is proved for all integer n >= 1.\r\n" );
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