![]() ![]() Here are two more examples of the numbers we seek: Fib(1)=1 and Fib(5)=5. Fib(11)=89 This time the digit sum is 8+9 = 17.īut 89 is not the 17th Fibonacci number, it is the 11th (its index number is 11) so the digit sum of 89 is not equal to its index number.Ĭan you find other Fibonacci numbers with a digit sum equal to its index number? So the index number of Fib(10) is equal to its digit sum. The sum of its digits is 5+5 or 10 and that is also the index number of 55 (10-th in the list of Fibonacci numbers). If we add all the digits of a number we get its digit sum.įind Fibonacci numbers for which the sum of the digits of Fib( n) is equal to its index number n:įor example:- Fib(10)=55 the tenth Fibonacci number is Fib(10) = 55. Digit Sums Michael Semprevivo suggests this investigation for you to try. for the last five digits the cycle length is 150,000.for the last four digits,the cycle length is 15,000 and.For the last three digits, the cycle length is 1,500.After Fib(300) the last two digits repeat the same sequence again and again. Take a look at the Fibonacci Numbers List or, better, see this list in another browser window, then you can refer to this page and the list together.Ġ, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 1 44, 2 33, 3 77, 6 10, 9 87. We can also make the Fibonacci numbers appear in a decimal fraction, introduce you to an easily learned number magic trick that only works with Fibonacci-like series numbers, see how Pythagoras' Theorem and right-angled triangles such as 3-4-5 have connections with the Fibonacci numbers and then give you lots of hints and suggestions for finding more number patterns of your own. We also relate Fibonacci numbers to Pascal's triangle via the original rabbit problem that Fibonacci used to introduce the series we now call by his name. There is an unexpected pattern in the initial digits too. So, we return the n-1 index element instead of the n index element.The Mathematical Magic of the Fibonacci Numbers The Mathematical Magic of the Fibonacci Numbers This page looks at some patterns in the Fibonacci numbers themselves, from the digits in the numbers to their factors and multiples and which are prime numbers. Hence, when we say ’nth Fibonacci number’ or ’nth element of the array’, the element holds the index value n-1. The i-th element’s value is equal to the sum of the previous two elements in that array.Īs the loop iterates, each value of the array is filled.įinally, after the loop terminates, the value with index n-1 is returned.Īs explained before, The array index value starts from 0, but we start counting from 1. Next, in the loop, we assign the value of each element of the array. As explained in the previous code, the loop will start with the value i=2 and end with the value i=n. Now, we take a for loop with the variable i in range(2,n+1). Hence, we leave the 1st number (index 0) and change the 2nd number (index 1) to the value 1. Hence, we give an additional space because the nth Fibonacci number will have the index value n+1.Īt the creation of the array, all values are ‘0’ by default. ![]() But, the index of an array starts from 0. We start by creating an array FibArray of size n+1 – This is because, when we say nth Fibonacci number’, we start counting from 1. In this method, an array of size n is created by repeated addition using the for loop. Once the loop is terminated, the function returns the value of b, which stores the value of the nth Fibonacci number.This process continues and value 3 keeps reassigning until the loop terminates.To put it more simply, after c becomes a+b, a = b and b = c. Subsequently, the value of b is reassigned to the value of c. Once c takes the value of a+b, the value of a is reassigned to b. Consider the series to be quite literally in the sequence of a, b, & c.This variable is used to store the sum of the previous two elements in the series. Over here, we take a storage variable c.This range function means the loop starts with the value 2 and keeps iterating until the value n-1. If n is greater than 2, we take a ‘for’ loop of i in range( 2,n). Then, we return 0 for input value n=1 and 1 for input value n=2 using if-else statements.We take 2 pre-assigned variables a=0 and b=1 – they are the 1st & 2nd elements of the series.However, dynamic programming uses recursion to achieve repeated addition, while this method uses the `for loop’. ![]() ![]() This method is almost entirely the same as dynamic programming. ![]()
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