# 51st Fibonacci number

Calculating the 51st 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 (51 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

51st Fibonacci number is: 20365011074.

## Sum of the first 51 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 = 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 = 53316291172

Sum of the first 51 Fibonacci numbers: 53316291172.
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