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You can put this solution on YOUR website!
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\n" ); document.write( "If a_1, a_2, ..., a_19 satisfy\r
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\n" ); document.write( "\n" ); document.write( "a_1 + a_2 + a_3 + a_4 = 1,
\n" ); document.write( "a_2 + a_3 + a_4 + a_5 = 2,
\n" ); document.write( "a_3 + a_4 + a_5 + a_6 = 3, \r
\n" ); document.write( "\n" ); document.write( "...\r
\n" ); document.write( "\n" ); document.write( "a_{16} + a_{17} + a_{18} + a_{19} = 16,
\n" ); document.write( "a_{17} + a_{18} + a_{19} + a_1 = 17,\r
\n" ); document.write( "\n" ); document.write( "a_{18} + a_{19} + a_1 + a_2 = 18,\r
\n" ); document.write( "\n" ); document.write( "a_{19} + a_1 + a_2 + a_3 = 19,\r
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\n" ); document.write( "\n" ); document.write( "what is the value of a_19?
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\n" ); document.write( "\n" ); document.write( "The solution and the final answer in the post by @CPhill are INCORRECT.\r
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\n" ); document.write( "\n" ); document.write( "The idea is good and the implementation is good until some point.
\n" ); document.write( "After this point, implementation is wrong.\r
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\n" ); document.write( "\n" ); document.write( "So, I will copy-paste the solution by @CPhill below in my post.
\n" ); document.write( "Then I will mark there the point till which the solution is correct.
\n" ); document.write( "After this point, I will place my calculations, and will complete the solution to the end.\r
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document.write( "Let's analyze the given system of equations.\r\n" );
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document.write( "We have:\r\n" );
document.write( "1. a_1 + a_2 + a_3 + a_4 = 1\r\n" );
document.write( "2. a_2 + a_3 + a_4 + a_5 = 2\r\n" );
document.write( "3. a_3 + a_4 + a_5 + a_6 = 3\r\n" );
document.write( "...\r\n" );
document.write( "16. a_16 + a_17 + a_18 + a_19 = 16\r\n" );
document.write( "17. a_17 + a_18 + a_19 + a_1 = 17\r\n" );
document.write( "18. a_18 + a_19 + a_1 + a_2 = 18\r\n" );
document.write( "19. a_19 + a_1 + a_2 + a_3 = 19\r\n" );
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document.write( "Subtract equation (1) from equation (2):\r\n" );
document.write( " (a_2 + a_3 + a_4 + a_5) - (a_1 + a_2 + a_3 + a_4) = 2 - 1\r\n" );
document.write( " a_5 - a_1 = 1\r\n" );
document.write( " a_5 = a_1 + 1\r\n" );
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document.write( "Subtract equation (2) from equation (3):\r\n" );
document.write( " (a_3 + a_4 + a_5 + a_6) - (a_2 + a_3 + a_4 + a_5) = 3 - 2\r\n" );
document.write( " a_6 - a_2 = 1\r\n" );
document.write( " a_6 = a_2 + 1\r\n" );
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document.write( "In general, we have a_{n+4} = a_n + 1.\r\n" );
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document.write( "This means:\r\n" );
document.write( " a_5 = a_1 + 1\r\n" );
document.write( " a_9 = a_5 + 1 = a_1 + 2\r\n" );
document.write( " a_13 = a_9 + 1 = a_1 + 3\r\n" );
document.write( " a_17 = a_13 + 1 = a_1 + 4\r\n" );
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document.write( "Similarly,\r\n" );
document.write( " a_6 = a_2 + 1\r\n" );
document.write( " a_10 = a_6 + 1 = a_2 + 2\r\n" );
document.write( " a_14 = a_10 + 1 = a_2 + 3\r\n" );
document.write( " a_18 = a_14 + 1 = a_2 + 4\r\n" );
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document.write( "Also,\r\n" );
document.write( " a_7 = a_3 + 1\r\n" );
document.write( " a_11 = a_7 + 1 = a_3 + 2\r\n" );
document.write( " a_15 = a_11 + 1 = a_3 + 3\r\n" );
document.write( " a_19 = a_15 + 1 = a_3 + 4\r\n" );
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document.write( "And,\r\n" );
document.write( " a_8 = a_4 + 1\r\n" );
document.write( " a_12 = a_8 + 1 = a_4 + 2\r\n" );
document.write( " a_16 = a_12 + 1 = a_4 + 3\r\n" );
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document.write( "Now let's use equations 1, 17, 18, and 19.\r\n" );
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document.write( "From eq. 1 --> a_1 + a_2 + a_3 + a_4 = 1\r\n" );
document.write( "From eq.17 --> a_17 + a_18 + a_19 + a_1 = 17\r\n" );
document.write( "From eq.18 --> a_18 + a_19 + a_1 + a_2 = 18\r\n" );
document.write( "From eq.19 --> a_19 + a_1 + a_2 + a_3 = 19 (*)\r\n" );
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document.write( "Substituting the expressions we found:\r\n" );
document.write( "17. (a_1 + 4) + (a_2 + 4) + (a_3 + 4) + a_1 = 17\r\n" );
document.write( " 2a_1 + a_2 + a_3 + 12 = 17\r\n" );
document.write( " 2a_1 + a_2 + a_3 = 5 (17)\r\n" );
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document.write( "18. (a_2 + 4) + (a_3 + 4) + a_1 + a_2 = 18\r\n" );
document.write( " a_1 + 2a_2 + a_3 + 8 = 18\r\n" );
document.write( " a_1 + 2a_2 + a_3 = 10 (18)\r\n" );
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document.write( "19. (a_3 + 4) + a_1 + a_2 + a_3 = 19\r\n" );
document.write( " a_1 + a_2 + 2a_3 + 4 = 19\r\n" );
document.write( " a_1 + a_2 + 2a_3 = 15 (19) \r\n" );
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document.write( " <<<---=== till this point, the solution by @CPhill is correct.\r\n" );
document.write( " After this point, it is WRONG.\r\n" );
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document.write( " <<<---=== So, starting from this point, my solution is going.\r\n" );
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document.write( "20. Add equations (17), (18) and (19). You will get\r\n" );
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document.write( " 4a_1 + 4a_2 + 4a_3 = 5 + 10 + 15 \r\n" );
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document.write( " 4a_1 + 4a_2 + 4a_3 = 30.\r\n" );
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document.write( " Divide both sides by 4\r\n" );
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document.write( " a_1 + a_2 + a_3 = 7.5. (20)\r\n" );
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document.write( "Now we are on the finish line.\r\n" );
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document.write( "From equation (17) subtract equation (20). You will get a_1 = 5 - 7.5 = -2.5.\r\n" );
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document.write( "From equation (18) subtract equation (20). You will get a_2 = 10 - 7.5 = 2.5.\r\n" );
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document.write( "From equation (19) subtract equation (20). You will get a_3 = 15 - 7.5 = 7.5.\r\n" );
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document.write( "From equation (*), a_19 = 19 - a_1 - a_2 - a_3 = 19 - 0 - (-2.5) - 2.5 - 7.5 = 11.5\r\n" );
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document.write( "For completeness, let's determine a_1 from the very first equation in this post\r\n" );
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document.write( " a_1 = 1 - (a_2 + a_3 + a_4) = 1 - ((-2.5) + 2.5 + 7.5) = 1 - 7.5 = -6.5.\r\n" );
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document.write( "From this point, all 19 terms a_1, a_2, a_3, a_4, a_5, . . . , a_19 can be determined.\r\n" );
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document.write( "See this table below\r\n" );
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document.write( " a_1 a_2 a_3 a_4 a_5 a_6 a_7 a_8 a_9 a_10 a_11 a_12 a_13 a_14 a_15 a_16 a_17 a_18 a_19 \r\n" );
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document.write( " -2.5 2.5 7.5 -6.5 -1.5 3.5 8.5 -5.5 -0.5 4.5 9.5 -4.5 0.5 5.5 10.5 5.5 1.5 6.5 11.5\r\n" );
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document.write( "You may check that all given 19 original equations are satisfied.\r\n" );
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document.write( "Final Answer: The final answer is a_19 = 11.5.\r\n" );
document.write( "\r\n" ); document.write( "\n" ); document.write( "Solved completely and correctly.\r
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\n" ); document.write( "\n" ); document.write( "By the way, I looked at the solution by Google AI for this problem of today, 03/02/25. The link to it is\r
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\n" ); document.write( "\n" ); document.write( "https://www.google.com/search?q=If+a_1%2C+a_2%2C+...%2C+a_19+satisfy+a_1+%2B+a_2+%2B+a_3+%2B+a_4+%3D+1%2C+a_2+%2B+a_3+%2B+a_4+%2B+a_5+%3D+2%2C+a_3+%2B+a_4+%2B+a_5+%2B+a_6+%3D+3%2C+...+a_%7B16%7D+%2B+a_%7B17%7D+%2B+a_%7B18%7D+%2B+a_%7B19%7D+%3D+16%2C+a_%7B17%7D+%2B+a_%7B18%7D+%2B+a_%7B19%7D+%2B+a_1+%3D+17%2C+a_%7B18%7D+%2B+a_%7B19%7D+%2B+a_1+%2B+a_2+%3D+18%2C+a_%7B19%7D+%2B+a_1+%2B+a_2+%2B+a_3+%3D+19%2C+what+is+the+value+of+a_19%3F&rlz=1C1CHBF_enUS1071US1071&oq=If+a_1%2C+a_2%2C+...%2C+a_19+satisfy+++a_1+%2B+a_2+%2B+a_3+%2B+a_4+%3D+1%2C++a_2+%2B+a_3+%2B+a_4+%2B+a_5+%3D+2%2C++a_3+%2B+a_4+%2B+a_5+%2B+a_6+%3D+3%2C+++...++a_%7B16%7D+%2B+a_%7B17%7D+%2B+a_%7B18%7D+%2B+a_%7B19%7D+%3D+16%2C+a_%7B17%7D+%2B+a_%7B18%7D+%2B+a_%7B19%7D+%2B+a_1+%3D+17%2C++a_%7B18%7D+%2B+a_%7B19%7D+%2B+a_1+%2B+a_2+%3D+18%2C++a_%7B19%7D+%2B+a_1+%2B+a_2+%2B+a_3+%3D+19%2C+++what+is+the+value+of+a_19%3F&gs_lcrp=EgZjaHJvbWUyBggAEEUYOdIBCTE5NjZqMGoxNagCCLACAfEFoMFf_ttl--Y&sourceid=chrome&ie=UTF-8\r
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\n" ); document.write( "\n" ); document.write( "This solution was slightly different from that by @CPhill, but also was WRONG.\r
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\n" ); document.write( "\n" ); document.write( "Naturally, I reported them about their wrong solution.\r
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\n" ); document.write( "\n" ); document.write( "Hope they will fix their artificial mind.\r
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