38th Fibonacci number

Calculating the 38th 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 (38 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

Sum of the first 38 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 = 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 = 102334154