The reciprocal Fibonacci constant is defined by the infinite sum: Psi = sigma_n=

Question

The reciprocal Fibonacci constant is defined by the infinite sum: Psi = sigma_n=1^infinity 1/F_n Where F_n are the Fibonacci numbers 1, 1, 2, 3, 5, 8, 13, .... Each element in this sequence of numbers is the sum of the previous two. Write a MATLAB program in a script file that calculates the first hundred Fibonacci numbers F_n. Start by setting the first two elements in F_n series equal to 1, then determine the series F_n up to n=100. Then find the value of the constant Psi for n = 10 to 100 with an increment of 10. The program must display the result using fprintf as: The Fibonacci constant for n = XXX is XXXXX The Fibonacci constant for n = XXX is XXXXX where XX is the value of n used and XXXXX is the value of the constant Psi displayed in f format with 15 decimal digits. What do you conclude from your results?

Explanation / Answer

%program to print 100 fibonacci numbers
N=100;
fib=zeros(1,N);
fib(1)=1;
fib(2)=1;
%printing first two fibonacci numbers
fprintf('The Fibonacci constant for n = %d is = %d ',1,fib(1));
fprintf('The Fibonacci constant for n = %d is = %d ',2,fib(2));
      
k=3;
%printing remaining fibonacci numbers
while k <= N
    fib(k)=fib(k-2)+fib(k-1);
    fprintf('The Fibonacci constant for n = %d is = %d ',k,fib(k));
    k=k+1;
end
fprintf(' ');

ouput:-

>> fibnocci
The Fibonacci constant for n = 1 is = 1
The Fibonacci constant for n = 2 is = 1
The Fibonacci constant for n = 3 is = 2
The Fibonacci constant for n = 4 is = 3
The Fibonacci constant for n = 5 is = 5
The Fibonacci constant for n = 6 is = 8
The Fibonacci constant for n = 7 is = 13
The Fibonacci constant for n = 8 is = 21
The Fibonacci constant for n = 9 is = 34
The Fibonacci constant for n = 10 is = 55
The Fibonacci constant for n = 11 is = 89
The Fibonacci constant for n = 12 is = 144
The Fibonacci constant for n = 13 is = 233
The Fibonacci constant for n = 14 is = 377
The Fibonacci constant for n = 15 is = 610
The Fibonacci constant for n = 16 is = 987
The Fibonacci constant for n = 17 is = 1597
The Fibonacci constant for n = 18 is = 2584
The Fibonacci constant for n = 19 is = 4181
The Fibonacci constant for n = 20 is = 6765
The Fibonacci constant for n = 21 is = 10946
The Fibonacci constant for n = 22 is = 17711
The Fibonacci constant for n = 23 is = 28657
The Fibonacci constant for n = 24 is = 46368
The Fibonacci constant for n = 25 is = 75025
The Fibonacci constant for n = 26 is = 121393
The Fibonacci constant for n = 27 is = 196418
The Fibonacci constant for n = 28 is = 317811
The Fibonacci constant for n = 29 is = 514229
The Fibonacci constant for n = 30 is = 832040
The Fibonacci constant for n = 31 is = 1346269
The Fibonacci constant for n = 32 is = 2178309
The Fibonacci constant for n = 33 is = 3524578
The Fibonacci constant for n = 34 is = 5702887
The Fibonacci constant for n = 35 is = 9227465
The Fibonacci constant for n = 36 is = 14930352
The Fibonacci constant for n = 37 is = 24157817
The Fibonacci constant for n = 38 is = 39088169
The Fibonacci constant for n = 39 is = 63245986
The Fibonacci constant for n = 40 is = 102334155
The Fibonacci constant for n = 41 is = 165580141
The Fibonacci constant for n = 42 is = 267914296
The Fibonacci constant for n = 43 is = 433494437
The Fibonacci constant for n = 44 is = 701408733
The Fibonacci constant for n = 45 is = 1134903170
The Fibonacci constant for n = 46 is = 1836311903
The Fibonacci constant for n = 47 is = 2971215073
The Fibonacci constant for n = 48 is = 4807526976
The Fibonacci constant for n = 49 is = 7778742049
The Fibonacci constant for n = 50 is = 12586269025
The Fibonacci constant for n = 51 is = 20365011074
The Fibonacci constant for n = 52 is = 32951280099
The Fibonacci constant for n = 53 is = 53316291173
The Fibonacci constant for n = 54 is = 86267571272
The Fibonacci constant for n = 55 is = 139583862445
The Fibonacci constant for n = 56 is = 225851433717
The Fibonacci constant for n = 57 is = 365435296162
The Fibonacci constant for n = 58 is = 591286729879
The Fibonacci constant for n = 59 is = 956722026041
The Fibonacci constant for n = 60 is = 1548008755920
The Fibonacci constant for n = 61 is = 2504730781961
The Fibonacci constant for n = 62 is = 4052739537881
The Fibonacci constant for n = 63 is = 6557470319842
The Fibonacci constant for n = 64 is = 10610209857723
The Fibonacci constant for n = 65 is = 17167680177565
The Fibonacci constant for n = 66 is = 27777890035288
The Fibonacci constant for n = 67 is = 44945570212853
The Fibonacci constant for n = 68 is = 72723460248141
The Fibonacci constant for n = 69 is = 117669030460994
The Fibonacci constant for n = 70 is = 190392490709135
The Fibonacci constant for n = 71 is = 308061521170129
The Fibonacci constant for n = 72 is = 498454011879264
The Fibonacci constant for n = 73 is = 806515533049393
The Fibonacci constant for n = 74 is = 1304969544928657
The Fibonacci constant for n = 75 is = 2111485077978050
The Fibonacci constant for n = 76 is = 3416454622906707
The Fibonacci constant for n = 77 is = 5527939700884758
The Fibonacci constant for n = 78 is = 8944394323791464
The Fibonacci constant for n = 79 is = 14472334024676220
The Fibonacci constant for n = 80 is = 23416728348467684
The Fibonacci constant for n = 81 is = 37889062373143904
The Fibonacci constant for n = 82 is = 61305790721611584
The Fibonacci constant for n = 83 is = 99194853094755488
The Fibonacci constant for n = 84 is = 160500643816367072
The Fibonacci constant for n = 85 is = 259695496911122560
The Fibonacci constant for n = 86 is = 420196140727489664
The Fibonacci constant for n = 87 is = 679891637638612224
The Fibonacci constant for n = 88 is = 1100087778366101888
The Fibonacci constant for n = 89 is = 1779979416004713984
The Fibonacci constant for n = 90 is = 2880067194370816000
The Fibonacci constant for n = 91 is = 4660046610375530496
The Fibonacci constant for n = 92 is = 7540113804746346496
The Fibonacci constant for n = 93 is = 1.220016e+19
The Fibonacci constant for n = 94 is = 1.974027e+19
The Fibonacci constant for n = 95 is = 3.194043e+19
The Fibonacci constant for n = 96 is = 5.168071e+19
The Fibonacci constant for n = 97 is = 8.362114e+19
The Fibonacci constant for n = 98 is = 1.353019e+20
The Fibonacci constant for n = 99 is = 2.189230e+20
The Fibonacci constant for n = 100 is = 3.542248e+20

upto 92 fibonacci numbers printted fine..

from 93 to 100 the fibonacci numbers are normalized

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