The Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, 21, ... starts with two 1s, and each term afterwards is the sum of its two predecessors. Which one of the ten digits is the last to appear in the units position of a number in the Fibonacci sequence?
just list the terms. 6 is the last to appear.
To determine which digit is the last to appear in the units position of a number in the Fibonacci sequence, we can find the remainders when each Fibonacci number is divided by 10.
Starting with the first few Fibonacci numbers:
F(1) = 1 % 10 = 1
F(2) = 1 % 10 = 1
F(3) = (F(1) + F(2)) % 10 = (1 + 1) % 10 = 2
F(4) = (F(2) + F(3)) % 10 = (1 + 2) % 10 = 3
F(5) = (F(3) + F(4)) % 10 = (2 + 3) % 10 = 5
F(6) = (F(4) + F(5)) % 10 = (3 + 5) % 10 = 8
F(7) = (F(5) + F(6)) % 10 = (5 + 8) % 10 = 3
F(8) = (F(6) + F(7)) % 10 = (8 + 3) % 10 = 1
F(9) = (F(7) + F(8)) % 10 = (3 + 1) % 10 = 4
F(10) = (F(8) + F(9)) % 10 = (1 + 4) % 10 = 5
As we can see, the Fibonacci sequence starts repeating after reaching F(60). Therefore, to find the last digit, we need to look at F(10), which is 5.
So, the digit 5 is the last one to appear in the units position of a number in the Fibonacci sequence.
To determine which digit is the last to appear in the units position of a number in the Fibonacci sequence, we can list out the Fibonacci numbers until we find a pattern or until we reach a point where all digits have appeared in the units position.
Let's start listing the Fibonacci sequence:
1, 1, 2, 3, 5, 8, 13, 21, ...
As we generate more terms in the sequence, we need to keep track of the units digit of each number.
1, 1, 2, 3, 5, 8, 3, 1, 4, 5, 9, 4, 3, 7, 0, 7, ...
At this point, we can see that digits 0, 1, 2, 3, 4, 5, 7, and 9 have already appeared in the units position. The only remaining digits are 6 and 8.
Let's continue listing more terms:
1, 1, 2, 3, 5, 8, 13, 21, 3, 24, 7, 1, 8, 9, 17, 6, 3, 9, 2, ...
Now, digit 6 has appeared, leaving only digit 8 remaining.
Continuing further, we will eventually see that digit 8 also appears:
1, 1, 2, 3, 5, 8, 13, 21, 3, 24, 7, 1, 8, 9, 17, 6, 3, 9, 2, 11, 13, ...
Hence, the digit 8 is the last digit to appear in the units position of a number in the Fibonacci sequence.