84th Fibonacci number

Calculating the 84th Fibonacci number without a calculator or programming language can be quite tedious due to the large numbers involved. However, if you're up for the challenge, you can use a method called "iterative Fibonacci sequence generation" which involves systematically generating each Fibonacci number in sequence until you reach the desired index.

Here's how you can do it manually:

Step 1: Initialize Variables

Start with two variables initialized to the first two Fibonacci numbers: F0 = 0 and F1 = 1.

Step 2: Iterate to Generate Fibonacci Numbers

Continue generating Fibonacci numbers iteratively until you reach the desired index (84 in this case). The next Fibonacci number in the sequence is the sum of the previous two Fibonacci numbers.

F2 = F1 + F0 = 1 + 0 = 1

F3 = F2 + F1 = 1 + 1 = 2

F4 = F3 + F2 = 2 + 1 = 3

F5 = F4 + F3 = 3 + 2 = 5

F6 = F5 + F4 = 5 + 3 = 8

F7 = F6 + F5 = 8 + 5 = 13

F8 = F7 + F6 = 13 + 8 = 21

F9 = F8 + F7 = 21 + 13 = 34

F10 = F9 + F8 = 34 + 21 = 55

F11 = F10 + F9 = 55 + 34 = 89

F12 = F11 + F10 = 89 + 55 = 144

F13 = F12 + F11 = 144 + 89 = 233

F14 = F13 + F12 = 233 + 144 = 377

F15 = F14 + F13 = 377 + 233 = 610

F16 = F15 + F14 = 610 + 377 = 987

F17 = F16 + F15 = 987 + 610 = 1597

F18 = F17 + F16 = 1597 + 987 = 2584

F19 = F18 + F17 = 2584 + 1597 = 4181

F20 = F19 + F18 = 4181 + 2584 = 6765

F21 = F20 + F19 = 6765 + 4181 = 10946

F22 = F21 + F20 = 10946 + 6765 = 17711

F23 = F22 + F21 = 17711 + 10946 = 28657

F24 = F23 + F22 = 28657 + 17711 = 46368

F25 = F24 + F23 = 46368 + 28657 = 75025

F26 = F25 + F24 = 75025 + 46368 = 121393

F27 = F26 + F25 = 121393 + 75025 = 196418

F28 = F27 + F26 = 196418 + 121393 = 317811

F29 = F28 + F27 = 317811 + 196418 = 514229

F30 = F29 + F28 = 514229 + 317811 = 832040

F31 = F30 + F29 = 832040 + 514229 = 1346269

F32 = F31 + F30 = 1346269 + 832040 = 2178309

F33 = F32 + F31 = 2178309 + 1346269 = 3524578

F34 = F33 + F32 = 3524578 + 2178309 = 5702887

F35 = F34 + F33 = 5702887 + 3524578 = 9227465

F36 = F35 + F34 = 9227465 + 5702887 = 14930352

F37 = F36 + F35 = 14930352 + 9227465 = 24157817

F38 = F37 + F36 = 24157817 + 14930352 = 39088169

F39 = F38 + F37 = 39088169 + 24157817 = 63245986

F40 = F39 + F38 = 63245986 + 39088169 = 102334155

F41 = F40 + F39 = 102334155 + 63245986 = 165580141

F42 = F41 + F40 = 165580141 + 102334155 = 267914296

F43 = F42 + F41 = 267914296 + 165580141 = 433494437

F44 = F43 + F42 = 433494437 + 267914296 = 701408733

F45 = F44 + F43 = 701408733 + 433494437 = 1134903170

F46 = F45 + F44 = 1134903170 + 701408733 = 1836311903

F47 = F46 + F45 = 1836311903 + 1134903170 = 2971215073

F48 = F47 + F46 = 2971215073 + 1836311903 = 4807526976

F49 = F48 + F47 = 4807526976 + 2971215073 = 7778742049

F50 = F49 + F48 = 7778742049 + 4807526976 = 12586269025

F51 = F50 + F49 = 12586269025 + 7778742049 = 20365011074

F52 = F51 + F50 = 20365011074 + 12586269025 = 32951280099

F53 = F52 + F51 = 32951280099 + 20365011074 = 53316291173

F54 = F53 + F52 = 53316291173 + 32951280099 = 86267571272

F55 = F54 + F53 = 86267571272 + 53316291173 = 139583862445

F56 = F55 + F54 = 139583862445 + 86267571272 = 225851433717

F57 = F56 + F55 = 225851433717 + 139583862445 = 365435296162

F58 = F57 + F56 = 365435296162 + 225851433717 = 591286729879

F59 = F58 + F57 = 591286729879 + 365435296162 = 956722026041

F60 = F59 + F58 = 956722026041 + 591286729879 = 1548008755920

F61 = F60 + F59 = 1548008755920 + 956722026041 = 2504730781961

F62 = F61 + F60 = 2504730781961 + 1548008755920 = 4052739537881

F63 = F62 + F61 = 4052739537881 + 2504730781961 = 6557470319842

F64 = F63 + F62 = 6557470319842 + 4052739537881 = 10610209857723

F65 = F64 + F63 = 10610209857723 + 6557470319842 = 17167680177565

F66 = F65 + F64 = 17167680177565 + 10610209857723 = 27777890035288

F67 = F66 + F65 = 27777890035288 + 17167680177565 = 44945570212853

F68 = F67 + F66 = 44945570212853 + 27777890035288 = 72723460248141

F69 = F68 + F67 = 72723460248141 + 44945570212853 = 117669030460994

F70 = F69 + F68 = 117669030460994 + 72723460248141 = 190392490709135

F71 = F70 + F69 = 190392490709135 + 117669030460994 = 308061521170129

F72 = F71 + F70 = 308061521170129 + 190392490709135 = 498454011879264

F73 = F72 + F71 = 498454011879264 + 308061521170129 = 806515533049393

F74 = F73 + F72 = 806515533049393 + 498454011879264 = 1304969544928657

F75 = F74 + F73 = 1304969544928657 + 806515533049393 = 2111485077978050

F76 = F75 + F74 = 2111485077978050 + 1304969544928657 = 3416454622906707

F77 = F76 + F75 = 3416454622906707 + 2111485077978050 = 5527939700884757

F78 = F77 + F76 = 5527939700884757 + 3416454622906707 = 8944394323791464

F79 = F78 + F77 = 8944394323791464 + 5527939700884757 = 14472334024676221

F80 = F79 + F78 = 14472334024676221 + 8944394323791464 = 23416728348467685

F81 = F80 + F79 = 23416728348467685 + 14472334024676221 = 37889062373143906

F82 = F81 + F80 = 37889062373143906 + 23416728348467685 = 61305790721611591

F83 = F82 + F81 = 61305790721611591 + 37889062373143906 = 99194853094755497

F84 = F83 + F82 = 99194853094755497 + 61305790721611591 = 160500643816367088

84th Fibonacci number is: 160500643816367088.

Sum of the first 84 Fibonacci numbers

SUMM = F1 + F2 + F3 + F4 + F5 + F6 + F7 + F8 + F9 + F10 + F11 + F12 + F13 + F14 + F15 + F16 + F17 + F18 + F19 + F20 + F21 + F22 + F23 + F24 + F25 + F26 + F27 + F28 + F29 + F30 + F31 + F32 + F33 + F34 + F35 + F36 + F37 + F38 + F39 + F40 + F41 + F42 + F43 + F44 + F45 + F46 + F47 + F48 + F49 + F50 + F51 + F52 + F53 + F54 + F55 + F56 + F57 + F58 + F59 + F60 + F61 + F62 + F63 + F64 + F65 + F66 + F67 + F68 + F69 + F70 + F71 + F72 + F73 + F74 + F75 + F76 + F77 + F78 + F79 + F80 + F81 + F82 + F83 + F84 = 1 + 1 + 2 + 3 + 5 + 8 + 13 + 21 + 34 + 55 + 89 + 144 + 233 + 377 + 610 + 987 + 1597 + 2584 + 4181 + 6765 + 10946 + 17711 + 28657 + 46368 + 75025 + 121393 + 196418 + 317811 + 514229 + 832040 + 1346269 + 2178309 + 3524578 + 5702887 + 9227465 + 14930352 + 24157817 + 39088169 + 63245986 + 102334155 + 165580141 + 267914296 + 433494437 + 701408733 + 1134903170 + 1836311903 + 2971215073 + 4807526976 + 7778742049 + 12586269025 + 20365011074 + 32951280099 + 53316291173 + 86267571272 + 139583862445 + 225851433717 + 365435296162 + 591286729879 + 956722026041 + 1548008755920 + 2504730781961 + 4052739537881 + 6557470319842 + 10610209857723 + 17167680177565 + 27777890035288 + 44945570212853 + 72723460248141 + 117669030460994 + 190392490709135 + 308061521170129 + 498454011879264 + 806515533049393 + 1304969544928657 + 2111485077978050 + 3416454622906707 + 5527939700884757 + 8944394323791464 + 14472334024676221 + 23416728348467685 + 37889062373143906 + 61305790721611591 + 99194853094755497 + 160500643816367088 = 420196140727489672

Sum of the first 84 Fibonacci numbers: 420196140727489672.
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