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